STABILITY OF CONSTRAINED CAPILLARY
INTERFACES
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Joshua Blake Bostwick
May 2011
c 2011 Joshua Blake Bostwick
°
STABILITY OF CONSTRAINED CAPILLARY INTERFACES
Joshua Blake Bostwick, Ph.D.
Cornell University 2011
Surface tension acting at an interface holds an underlying liquid in motionless
equilibrium. If the interface is partially supported, the extent and type of wettingcontact can significantly influence the stability of the equilibrium configuration.
This dissertation addresses a number of questions related to the stability and
dynamics of constrained interfaces subject to capillarity. The problems considered
here fall into two distinct classes.
The first class of problem is concerned with the extent-of-constraint and restricts to interface disturbances with immobile or ‘pinned’ contact lines. Here the
base-state volume is taken to be ‘filling’, or the static interface shape intersects
the surface-of-support tangentially, such as a spherical-belt or cylindrical-cup solid
support. With regards to the drop under spherical-belt support, the base-state is
stable, oscillations persist and the effect of the constraint is to modify the spectrum and modal structure of the unconstrained problem, as well as introduce a
low-frequency mode. In contrast, a cylindrical interface is subject to capillary instability and more particularly Plateau-Rayleigh break-up, which can be partially
stabilized by either a cylindrical-cup or end-plane support. Here introducing a
weak-secondary curvature to the cylindrical cup-like support can either stabilize
or destabilize this new family of nearly-cylindrical toroids. This is referred to as
the torus-lift of the cylinder and is used in conjunction with a symmetrization procedure to prove a large-amplitude stability result for cylindrical lens-like menisci.
The second class of problem is focused on the dynamics of the three-phase
contact-line and also the effect of base-state volume. In considering the contactline dynamics of the sessile drop, an instability related to translational or ‘walking’
motion is catalogued and two types of modal crossings are observed. The first is
related to contact-line mobility and relevant to experimentally-observed contactline instabilities, while the second is associated with symmetry-breaking of the
hemispherical base-state. A similar ‘walking’ instability is observed for the static
rivulet, which is shown to be related to the phenomenon of rivulet meandering.
Lastly, a ‘bounding’ method is developed and applied to the family of catenoids
to reproduce well-known stability results and generate new ones without explicitly
calculating the second variation.
BIOGRAPHICAL SKETCH
Joshua Blake Bostwick was born on January 9th, 1981 in Wauwatosa, Wisconsin,
where he also graduated from high school in May 1999. As a youth, he played a
number of sports such as tennis, basketball and soccer. He enjoyed the most success
with soccer, traveling throughout the country with his club team to compete in
high-level tournaments and winning the Wisconsin State Championship in 1998
with his high school team.
In the fall of 1999, he enrolled at the University of Wisconsin-Milwaukee
(UWM) to pursue degrees in both Civil Engineering & Mechanics and Physics.
While at UWM, Josh had the privilege to work with Dr. Michael Weinert as an
REU student on first principles electronic structure calculations with application
to MgO(111) crystal defects. Around this time he was also introduced to Dr.
Chris Papadopoulos, who was the instructor in his Strength of Materials course.
Dr. Papadopoulos noticed Josh’s potential immediately and asked him to begin
grading papers for his mechanics classes. While working together, Dr. Papadopoulos introduced him to the more subtle points of mechanics. This sparked Josh’s
interest in the subject and ultimately led to the selection of his senior project on
Kane’s equations. On the basis of this work and also academic performance, he
was granted the 2005 Outstanding Student Award. Josh graduated from UWM
with a B.S.E. in Civil Engineering & Mechanics and a B.A. in Physics with a minor
in Mathematics in 2005.
During the fall of 2005, Josh began working on his Ph.D. in Theoretical & Applied Mechanics (TAM) at Cornell University. He worked as a teaching assistant
while completing his graduate coursework. In honor of his teaching accomplishments at Cornell, he was given the 2010 H.D. Block Award for teaching excellence
in engineering mathematics, which was awarded on the basis of both student and
iii
faculty feedback.
It was during his second year that he began working with Dr. Paul Steen on
the dynamics of constrained capillary interfaces with application to liquid drops,
bridges and toroids. During the fall semester of 2007, he had the opportunity to
work, under a NSF IREE grant, on computational level set methods applied to
capillary interfaces with Dr. Peter Ehrhard at Dortmund Universitat in Dortmund,
Germany. The work completed under the direction of Dr. Steen, which has been
partially supported by a NASA grant, is disseminated in this dissertation and also
appears in select journal articles.
iv
I would like to dedicate this dissertation to Emily, for her patience and support.
v
ACKNOWLEDGEMENTS
A number of individuals have helped make this dissertation possible and I would
like to acknowledge a few of them here.
Foremost, I would like to express my deepest appreciation to my advisor, Paul
Steen, for his guidance and encouragement. It has been an honor and a privilege
to work with Paul. I would also like to recognize the other distinguished members
of my committee, Eberhard Bodenschatz (Physics) and Sidney Leibovich (MAE),
and thank them for providing valuable suggestions in preparing this thesis and
also taking the time to serve. In addition, I am grateful to the Department of
Theoretical & Applied Mechanics (TAM) for the learning environment they have
created and also their support during my tenure there. Finally, I would like to
thank Chris Papadopoulos for introducing me to the field of Mechanics and TAM.
I am indebted to all the members of the Steen research group for many stimulating discussions and also their invaluable suggestions in preparing the content
of this dissertation. It has been a pleasure to work with so many talented people,
who share similar research interests.
It would not have been possible to complete this thesis work without the infinite
support of a loving family. I thank my mother Mary for her encouragement,
understanding and strength during hard times; my sister Audra for teaching me
about perseverance and redemption; my niece Cora for bringing a little levity to
my day and reminding me not to take life/myself so seriously; and finally my late
father Ed, who has always driven me to succeed and continues to inspire me daily.
Lastly, in numerical order only, I would like to express my gratitude to Emily
Jung, whose sacrifices have more than paralleled my own throughout this journey.
It has truly been a blessing to have her in my life.
vi
TABLE OF CONTENTS
Biographical Sketch
Dedication . . . . .
Acknowledgements
Table of Contents .
List of Figures . . .
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. iii
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. vi
. vii
. xi
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Mathematical background . . . . . . . . . . . . . . . . . . . . . . .
1.3 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
4
10
2 Oscillations of a pinned liquid drop
2.1 Introduction . . . . . . . . . . . . . . . . . . . .
2.2 Mathematical formulation . . . . . . . . . . . .
2.2.1 Dynamic equations . . . . . . . . . . . .
2.3 Reduced system . . . . . . . . . . . . . . . . . .
2.3.1 Velocity potential solution . . . . . . . .
2.3.2 Surface deformation . . . . . . . . . . .
2.4 Solution method . . . . . . . . . . . . . . . . .
2.4.1 Constrained function space . . . . . . . .
2.4.2 Reduction to matrix form . . . . . . . .
2.5 Results . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Decomposition of eigenmodes . . . . . .
2.5.2 Center-of-mass motion . . . . . . . . . .
2.5.3 Extension to a double pinned fluid drop
2.5.4 Density variation . . . . . . . . . . . . .
2.6 Concluding remarks . . . . . . . . . . . . . . . .
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14
14
18
19
20
21
22
23
24
25
27
30
32
35
38
40
3 Viscous oscillations of a fluid drop under spherical-belt constraint
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Velocity field definition . . . . . . . . . . . . . . . . . . . . .
3.2.3 Reduced system . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Boundary/Integral conditions . . . . . . . . . . . . . . . . .
3.2.5 Velocity field decomposition . . . . . . . . . . . . . . . . . .
3.2.6 Velocity field governing equations . . . . . . . . . . . . . . .
3.3 Inviscid solution method . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Velocity potential solution . . . . . . . . . . . . . . . . . . .
3.3.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Operator equation . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Rayleigh-Ritz method . . . . . . . . . . . . . . . . . . . . .
44
44
48
50
50
51
51
52
53
54
55
56
56
57
vii
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3.4
3.5
3.6
3.7
3.3.5 Constrained function space . .
Inviscid results . . . . . . . . . . . .
3.4.1 Perturbed-volume embedding
Viscous solution method . . . . . . .
3.5.1 Vector potential solution . . .
3.5.2 Velocity potential solution . .
3.5.3 Pressure . . . . . . . . . . . .
3.5.4 Boundary conditions . . . . .
3.5.5 Viscous operator equation . .
3.5.6 Viscous operator solution . . .
Viscous results . . . . . . . . . . . .
3.6.1 Checks on viscous solution . .
Concluding remarks . . . . . . . . . .
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58
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69
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78
81
4 Stability of constrained cylindrical interfaces and the torus-lift of
Plateau-Rayleigh
84
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Formulation of dynamical problem . . . . . . . . . . . . . . . . . . 89
4.2.1 Hydrodynamic equations . . . . . . . . . . . . . . . . . . . . 91
4.2.2 Normal-mode reduction . . . . . . . . . . . . . . . . . . . . 92
4.2.3 Reduction to operator equation . . . . . . . . . . . . . . . . 93
4.2.4 Solution of operator equations . . . . . . . . . . . . . . . . . 95
4.3 Results for a constrained cylindrical interface . . . . . . . . . . . . . 98
4.4 Lifting the cylinder to the torus . . . . . . . . . . . . . . . . . . . . 102
4.4.1 Near-toroidal equilibrium shapes . . . . . . . . . . . . . . . 104
4.4.2 Stability of near-toroidal equilibrium shapes . . . . . . . . . 105
4.4.3 Symmetrization and large-amplitude stability of the lens
meniscus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Stability of the sessile drop: contact-line dynamics
breaking
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 Mathematical formulation . . . . . . . . . . . . . .
5.2.1 Hydrodynamic field equations . . . . . . . .
5.3 Derivation of functional eigenvalue equation . . . .
5.3.1 Normal-mode reduction . . . . . . . . . . .
5.3.2 Operator formalism . . . . . . . . . . . . . .
5.3.3 Contact-line conditions . . . . . . . . . . . .
5.4 Solution method for kinematic disturbances . . . .
5.4.1 Inverse-operator construction . . . . . . . .
5.4.2 Rayleigh-Ritz . . . . . . . . . . . . . . . . .
5.4.3 Results for kinematic disturbances . . . . .
5.5 Solution method for contact-line speed relation . . .
viii
and symmetry
115
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. . . . . . . . . 156
5.6
5.5.1 Operator construction
5.5.2 Hybrid Ritz method .
5.5.3 Results . . . . . . . . .
Concluding remarks . . . . . .
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6 Capillary instabilities of the static rivulet:
modes
6.1 Introduction . . . . . . . . . . . . . . . . . .
6.2 Mathematical formulation . . . . . . . . . .
6.2.1 Field equations . . . . . . . . . . . .
6.2.2 Normal-mode reduction . . . . . . .
6.2.3 Operator formalism . . . . . . . . . .
6.2.4 Contact-line conditions . . . . . . . .
6.2.5 Green’s function construction . . . .
6.2.6 Solution of operator equation . . . .
6.3 Results . . . . . . . . . . . . . . . . . . . . .
6.3.1 Static stability . . . . . . . . . . . .
6.3.2 Critical disturbance . . . . . . . . . .
6.4 Concluding Remarks . . . . . . . . . . . . .
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156
157
158
175
varicose and sinuous
181
. . . . . . . . . . . . . 181
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. . . . . . . . . . . . . 187
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. . . . . . . . . . . . . 196
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. . . . . . . . . . . . . 198
. . . . . . . . . . . . . 201
7 Stability bounds for the catenoid
7.1 Introduction . . . . . . . . . . . . . . . . . . . .
7.2 Static stability . . . . . . . . . . . . . . . . . .
7.2.1 Pinned contact-line disturbance . . . . .
7.2.2 Natural disturbance . . . . . . . . . . . .
7.2.3 Remarks . . . . . . . . . . . . . . . . . .
7.3 Dynamic stability . . . . . . . . . . . . . . . . .
7.3.1 Hydrodynamic formulation . . . . . . . .
7.3.2 Normal-mode reduction . . . . . . . . .
7.3.3 Solution of operator equation . . . . . .
7.3.4 Results . . . . . . . . . . . . . . . . . . .
7.4 Static stability of the axisymmetric liquid bridge
7.4.1 Plateau-Rayleigh instability . . . . . . .
7.4.2 Remarks . . . . . . . . . . . . . . . . . .
7.5 Concluding remarks . . . . . . . . . . . . . . . .
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203
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. 229
8 Future work
232
8.1 ‘Walking’ instability . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.2 Forced oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.3 Contact-line speed condition . . . . . . . . . . . . . . . . . . . . . . 234
A Linearization of Young-Dupré equation
ix
235
B Modified boundary condition calculations for the immiscible viscous drop
238
B.1 Modified boundary conditions . . . . . . . . . . . . . . . . . . . . . 238
B.2 Immiscible drop operator equation . . . . . . . . . . . . . . . . . . 239
C Rotational wave solution of the viscous drop under spherical-belt
constraint
240
D Constrained variational principle
242
D.1 Coupled non-linear springs . . . . . . . . . . . . . . . . . . . . . . . 250
x
LIST OF FIGURES
1.1
1.2
1.3
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
3.1
3.2
3.3
3.4
3.5
Three-phase contact-line: (a) definition sketch and (b) force balance
on contact-line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Disturbance classes: (a) fixed contact-angle (natural) and (b) fixed
contact-line (pinned). . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of an arbitrary fluid domain bound by a free surface ∂Df
and surface-of-support ∂Ds . . . . . . . . . . . . . . . . . . . . . . .
5
7
8
Definition sketch for the pinned liquid drop. . . . . . . . . . . . . .
Eigenfrequency against pin location (ρe /ρi = 0). . . . . . . . . . . .
Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = −0.99.
Mode n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ). . . . . . . . . . . .
Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = 0. Mode
n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ). . . . . . . . . . . . . . .
Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = −0.557.
Mode n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ). . . . . . . . . . . .
Center-of-mass motion contribution to eigenmodes as a function of
pin location ζ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency against second pin location ζ2 , while holding the first
pin location (a) ζ1 = 0.01 and (b) ζ1 = 0.15 fixed. . . . . . . . . . .
Frequency against second pin location ζ2 , while holding the first
pin location (a) ζ1 = −0.5 and (b) ζ1 = −0.7 fixed. . . . . . . . . .
Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −.55, ζ2 = 0.55. . . . . . . . . . . . . . . . . . . . . . . . . . .
Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −0.775, ζ2 = 0.001. . . . . . . . . . . . . . . . . . . . . . . . .
Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −0.91, ζ2 = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . .
Density variation in eigenmodes for a drop pinned on the (a) south
pole ζ = −0.99 and (b) equator ζ = 0. . . . . . . . . . . . . . . . .
Frequency versus pin location: Spherical cap (SC) and n = 1 mode.
Comparison with experiment (a) experiment and (b) n = 2 eigenmode for ζ = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
28
Definition sketch of a drop constrained by a spherical-belt. . . . .
Frequency (λ2 ) comparison between continuous (C), discontinuous
(DC) and spherical cap (SC) perturbations for a pinned circle-ofcontact (ζ1 = ζ2 ) with (a) n = 1 (b) n = 2, 3 . . . . . . . . . . . . .
Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 for a drop with pinned
circle-of-contact located at the equator (ζ1 = ζ2 = 0). . . . . . . . .
Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 with spherical-belt
constraint ζ1 = −0.7, ζ2 = 0. . . . . . . . . . . . . . . . . . . . . . .
Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 with spherical-belt
constraint ζ1 = −0.6, ζ2 = 0.8. . . . . . . . . . . . . . . . . . . . . .
49
xi
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30
31
35
37
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3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Frequency λ2 against second pin location ζ2 , while holding ζ1 = 0.4.
Mode shape (n = 3) at (a) point A(ζ1 = 0.4, ζ2 = 0.45) (b) point
B(ζ1 = 0.4, ζ2 = 0.85) (c) point C(ζ1 = 0.4, ζ2 = 0.95) of Figure 3.6.
Geometric index (engineering) of mode (a) n = 1, (b) n = 2, (c)
n = 3 and (d) n = 4, against ζ1 and ζ2 . . . . . . . . . . . . . . . .
Geometric index: comparison between the engineering and mathematical interpretation. . . . . . . . . . . . . . . . . . . . . . . . .
Perturbed volume Vn (a, b) and frequency λ2 (c, d) vs. ζ2 for fixed
(a),(c) ζ1 = 0.2 and (b),(d) ζ1 = 0.4. . . . . . . . . . . . . . . . . .
Indicator function Γ(x, ζ1 , ζ2 ) . . . . . . . . . . . . . . . . . . . . .
Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ] against
viscosity parameter ²i for a viscous drop with spherical-bowl support (ζ1 = −1, ζ2 = −0.8). . . . . . . . . . . . . . . . . . . . . . . .
Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ] against
viscosity parameter ²e for a bubble pinned at the south pole (ζ1 =
−1, ζ2 = −0.99). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ] against
viscosity parameter ²i for a drop with spherical-belt support (ζ1 =
−0.2, ζ2 = 0.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ] against
viscosity parameter ²e for a bubble with spherical-belt support
(ζ1 = −0.2, ζ2 = 0.4). . . . . . . . . . . . . . . . . . . . . . . . . . .
Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 for a drop with
spherical-belt support (ζ1 = −0.2, ζ2 = 0.4). . . . . . . . . . . . . .
Constrained cylindrical interface definition sketch: (a) cross-section
with solid support (thick) (b) axial-section with interface pinned at
endpoints and (c) 3D view, with sample disturbed interface. . . . .
Sample orthonormal basis function ψ(θ) . . . . . . . . . . . . . . .
Growth rate of most unstable polar mode [1, k] vs. aspect ratio
L/R (a) for θs = 0.01 with natural conditions and (b) for θs = 2.0
for pinned conditions. . . . . . . . . . . . . . . . . . . . . . . . . .
Dispersion relation of l = 1 modes (a) for θs = 0.01 with natural
and pinned conditions and (b) for varying θs for natural conditions.
Rayleigh dispersion, for reference. . . . . . . . . . . . . . . . . . . .
Static stability against polar constraint θs measured by (a) wavenumber αc or (b) envelope of stable L/R (below curve). . . . . . .
(a) Growth rate λ2m and (b) wave-number αm against polar constraint for fastest growing mode. . . . . . . . . . . . . . . . . . . .
Modes [l, k] in 3D and polar projection for L/R = 2π with (a, b)
θs = 2.0 and pinned conditions for (a) [1,1] and (b) [3,2] and (c, d)
θs = 0.01 and natural conditions for (c) [1,1] and (d) [2,3]. . . . .
Torus sketch in (a) 3D view and in (b) polar view with cup support
(thick line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
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68
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79
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89
97
98
99
100
100
101
103
4.9
Torus equilibrium shape r0 (θ) with unit-circle (dotted) for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Stability of toroidal equilibrium shapes showing (a) static limit αc
against deviation from cylinder ² and (b) typical destabilizing mode
shape, with equilibrium shape (dotted) for reference. . . . . . . . .
4.11 Symmetrization of general shape proceeds from (a) non-circular
slice AA0 to (b) circular slice BB0 with reassembly to axisymmetric
shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Symmetrization of lens meniscus using (a) parallel and transverse
cross-sectional slices in 3D view; sketches of large-amplitude disturbances that are (b) admissible (in transverse section), or (c, d)
inadmissible (c) in transverse section, (d) in parallel section. . . . .
Typical experimental relationship between contact-angle α and
contact-line speed uCL , which also shows the advancing αa and
receding αr static contact-angle (uCL → 0). . . . . . . . . . . . . .
5.2 Sessile drop equilibrium shape in polar view showing vectors normal
n and tangential t to the surface Γ. . . . . . . . . . . . . . . . . .
5.3 Definition sketch with unperturbed Γ (dashed) and perturbed interface η (solid) in (a) polar cross-section and three-dimensional (b)
perspective and (c) top views. . . . . . . . . . . . . . . . . . . . . .
5.4 Kinematic disturbances classes for the sessile drop: (a) natural and
(b) pinned contact-line. . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Typical contact-line behavior observed (solid) and the continuous
constitutive law model (dashed) imposed here. . . . . . . . . . . .
5.6 Frequency λk,l against contact-angle α for natural (Ang) and pinned
(Pin) modes [k, l] with low azimuthal wave-number l = 0, 1: (a)
[1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f ) [6, 0]. Here
Im[λ1,1 ] = 0 in all cases except α > 90◦ (c.f. figure 5.10). Note that
the scalings of the frequency axis are different from (a) to (f ). . . .
5.7 Pinned mode shapes [k, l] for hemispherical base-state α = 90◦ : (a)
[1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f ) [6, 0] (polar
view). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Natural mode shapes [k, l] for sub-hemispherical base-state α =
60◦ : (a) [1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f ) [6, 0]
(polar view). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Contact-line mobility of the natural mode shape [k, l] = [5, 1] for
contact-angle (a) 60◦ , (b) 90◦ , and (c) 120◦ . . . . . . . . . . . . . .
5.10 Instability growth rate −λ21,1 against contact-angle α for a sessile drop subject to a natural disturbance, exhibiting a maximum
growth rate (−λ21,1 = 0.0458) at α = 132.5◦ . . . . . . . . . . . . .
5.11 Typical instability mode shape [1, 1] with contact-angle α = 120◦
in (a) polar and (b) three-dimensional views. . . . . . . . . . . . .
104
106
107
111
5.1
xiii
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124
125
132
133
142
143
143
144
146
146
5.12 Instability mode shape [1, 1] with contact-angle (a) 91◦ , (b) 105◦ ,
(c) 120◦ , (d) 135◦ , (e) 150◦ , and (f ) 170◦ in polar view. . . . . . . .
5.13 Time series of the instability mode shape with largest growth rate
(−λ21,1 = 0.0458): (a, b, c) polar view and (d, e, f ) contact-line
footprint at time (a, d) t = 0, (b, e) t = T /2 and (c, f ) t = T
(α = 132.5◦ ). Here T represents the time it takes to disturb the
interface by a given amplitude (T = 1.226 in this figure). . . . . .
5.14 Decomposition of disturbance energy (E) against contact-angle α
for the [1, 1] instability mode. . . . . . . . . . . . . . . . . . . . .
5.15 Visualization of the left (Fl ) and right (Fr ) constraint force at the
contact-line for the natural mode shapes with (a) even and (b) odd
symmetry about the vertical mid-plane. . . . . . . . . . . . . . . .
5.16 Natural mode shapes [k, l] in polar and three-dimensional side/top
views for (a) zonal [8, 0] , (b) sectoral [5, 5] , and (c) tesseral [7, 3]
disturbances for α = 90◦ . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Pinned mode shapes [k, l] of the hemispherical base-state α = 90◦
in polar and three-dimensional side/top views for (a) zonal [4, 0] ,
(b) sectoral [3, 3] , and (c) tesseral [5, 1] disturbances. . . . . . . . .
5.18 Natural frequency λk,l , as it depends upon azimuthal wave-number
l, against contact-angle α for fixed polar wave-number (a) k = 4,
(b) k = 5, (c) k = 6, (d) k = 7, (e) k = 8, and (f ) k = 9. . . . . . .
5.19 Pinned frequency λk,l , as it depends upon azimuthal wave-number
l, against contact-angle α for fixed polar wave-number (a) k = 4,
(b) k = 5, (c) k = 6, (d) k = 7, (e) k = 8, and (f ) k = 9. . . . . . .
5.20 Oscillation frequency ωk,0 (a, d, g), decay rate γk,0 (b, e, h) and effective dissipation Qk,0 (c, f, i) as a function of the spreading parameter Λ for a contact-angle (a, b, c) α = 75◦ , (d, e, f ) α = 90◦ and
(g, h, i) α = 105◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21 Oscillation frequency ωk,3 (a), decay rate γk,3 (b) and effective dissipation Qk,3 (c) as a function of the spreading parameter Λ for
contact-angle α = 75◦ and azimuthal wave-number l = 3. . . . . . .
5.22 Oscillation frequency ω7,l (a), decay rate γ7,l (b) and effective dissipation Q7,l (c) as a function of the spreading parameter Λ for
contact-angle α = 75◦ and polar wave-number k = 7. . . . . . . . .
5.23 Oscillation frequency ω6,l (a), decay rate γ6,l (b) and effective dissipation Q6,l (c) as a function of the spreading parameter Λ for
contact-angle α = 105◦ and polar wave-number k = 6. . . . . . . .
5.24 Stability diagram for the [k, l] = [1, 1] mode: critical spreading
parameter Λc against contact-angle α. . . . . . . . . . . . . . . . .
5.25 Complex frequency λ = −γ + iω for the [1, 1] mode: (a) decay rate
γ1,1 > 0 and (b) instability growth rate γ1,1 < 0 as a function of the
spreading parameter Λ for contact-angle α = 105◦ . See figure 5.27
(a) for corresponding oscillation frequency ω1,1 . . . . . . . . . . .
xiv
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150
152
153
154
155
159
161
161
162
163
164
5.26 Oscillation frequency ωk,l , as it depends upon azimuthal wavenumber l, against spreading parameter Λ for the hemispherical
(α = 90◦ ) base-state with polar wave-number (a) k = 1, (b) k = 2,
(c) k = 3, (d) k = 4, (e) k = 5, and (f ) k = 6. . . . . . . . . . . . . 166
5.27 Oscillation frequency ωk,l , as it depends upon azimuthal wavenumber l, against spreading parameter Λ for a typical superhemispherical (α = 105◦ ) base-state with polar wave-number (a)
k = 1, (b) k = 2, (c) k = 3, (d) k = 4, (e) k = 5, and (f ) k = 6. . . 167
5.28 Modal crossings from the dynamic contact-line condition: oscillation frequency ωk,l as measured by the spreading parameter Λ for
α = 105◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.29 Modal crossings controlled by base-state volume: frequency λk,l as
a function of static contact-angle α for (a) natural and (b) pinned
disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.30 Spectral lines: (a) frequencies ωk,l , as they depend upon azimuthal
wave-number l, against polar wave-number k and (b) blow-up of
polar wave-numbers k = 5, 6 with natural mode shapes for α = 60◦ . 172
5.31 Spectral lines: (a) frequencies ωk,l , as they depend upon azimuthal
wave-number l, against polar wave-number k and (b) blow-up of
polar wave-numbers k = 7, 8 with pinned mode shapes for α = 120◦ . 172
5.32 Periodic table of mode shapes: filling order for (a) subhemispherical drop (α = 60◦ ) subject to a natural disturbance and
(b) super-hemispherical drop (α = 120◦ ) with pinned contact-lines. 174
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Definition sketch of the rivulet in (a) polar and (b) threedimensional perspective views. . . . . . . . . . . . . . . . . . . . . 186
Natural disturbances: polar view of the (a) varicose and (b) sinuous
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Pinned contact-line disturbances: polar view of the (a) varicose and
(b) sinuous modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Top view of typical three-dimensional (a) varicose and (b) sinuous
mode shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Dispersion relations: growth rate λ2 vs. axial wave-number k for
(a) varicose and (b) sinuous modes, subject to a natural disturbance.196
Static stability boundary ks against contact-angle α for (a) varicose and (b) sinuous modes for the natural (Ang) and pinned (Pin)
disturbances. Here stable (S) and unstable (U) regions are noted. 197
Fastest growing varicose mode: (a) wave-number km and (b) growth
rate λ2m against contact-angle α for natural (Ang) and pinned (Pin)
disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Typical instability mode shapes. The varicose mode for the (a)
natural and (b) pinned contact-line disturbance. The natural sinuous mode with a (c) polar (k = 0) and (d) typical axial (k 6= 0)
disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
xv
6.9
Fastest growing sinuous mode: (a) wave-number km and (b) growth
rate λ2m against contact-angle α for the natural disturbance. . . . . 200
6.10 Comparison between the varicose mode with pinned contact-lines
contacting either a cylindrical or planar support: maximum growth
rate λ2m against contact-angle α. . . . . . . . . . . . . . . . . . . . 200
7.1
7.2
Definition sketch for the catenoid. . . . . . . . . . . . . . . . . . . 207
Stability diagram for the pinned contact-line disturbance class
showing stable (solid line), unstable (dashed line) and conditionally
stable (dotted) catenoids. Here the conditionally stable catenoid is
stable (S) to constant-volume (Vol) and unstable (U) to constantpressure (Press) disturbances. . . . . . . . . . . . . . . . . . . . . 211
7.3 Scaling of catenoid by contact-angle: parameters (a) S and (b) c
against contact-angle α. . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4 Stability diagram for the catenoid subject to the natural disturbance class showing unstable (dashed line) and conditionally stable (solid line) regions. The conditionally stable catenoid is stable
(S) to constant-volume (Vol) and unstable (U) to constant-pressure
(Press) disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.5 Constant-volume stability diagram showing stable (solid line), unstable (dashed line) and conditionally stable (dotted line) catenoids.
The conditionally stable catenoid is unstable (U) to natural (Ang)
and stable (S) to pinned (Pin) disturbances. . . . . . . . . . . . . 215
7.6 Constant-pressure stability diagram showing unstable (dashed line)
and conditionally stable (solid line) catenoids. The conditionally
stable catenoid is unstable (U) to natural (Ang) and stable (S) to
pinned (Pin) disturbances. . . . . . . . . . . . . . . . . . . . . . . 216
7.7 Unstable growth rate λ21 < 0 vs. (a) slenderness Λ and (b) contactangle α for the catenoid given a natural disturbance. . . . . . . . . 224
7.8 Typical instability mode shape for the natural disturbance (α = 20◦ ).224
7.9 Oscillation mode (n = 2): (a) Frequency λ22 > 0 vs. contact-angle
α and (b) sample mode shape for α = 25◦ . . . . . . . . . . . . . . . 225
7.10 Oscillation mode (n = 3): (a) Frequency λ23 > 0 vs. contact-angle
α and (b) sample mode shape for α = 20◦ . . . . . . . . . . . . . . . 225
D.1
D.2
D.3
D.4
Local energy landscape for (a) E(0, 2) = x21 + x22 (Minimum), (b)
E(1, 1) = x21 − x22 (Saddle), and (c) E(2, 0) = −x21 − x22 (Maximum)
Contour plot of the energy landscape for (a) E(0, 2) = x21 + x22 , (b)
E(1, 1) = x21 − x22 , and (c) E(2, 0) = −x21 − x22 . . . . . . . . . . . .
Projection of the energy landscape for a saddle point showing slices
of (a) constant x2 (constrained minimum) and (b) constant x1 (constrained maximum) . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the system of coupled non-linear springs . . . . . . .
xvi
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244
245
251
D.5
D.6
Solution surface (a) F (x∗1 ), (b) F (x∗2 ), (c) F (x∗3 ) and (d) F (x∗4 )
against Lagrange multipliers µ1 , µ2 , where gray shading indicates a
constrained minimum. . . . . . . . . . . . . . . . . . . . . . . . . 254
Stability diagram for extremal (a) x∗1 , (b) x∗2 , (c) x∗3 and (d) x∗4
against Lagrange multipliers µ1 , µ2 . Gray shading indicates the
extremal is a constrained local minimum. . . . . . . . . . . . . . . 255
xvii
CHAPTER 1
INTRODUCTION
Capillary interfaces are ubiquitous in nature and have been studied extensively
since the time of Young (1805) and Laplace (1806), who introduced the notion
of surface tension. Early work on the subject was primarily focused on isolated
surfaces, while more recently there has been an interest in partially-supported
interfaces with an emphasis on the dynamics of the three-phase contact-line. Interest in wetting conditions can be attributed to a large number of capillary-based
technological innovations, such as liquid lens focusing, spin-casting processes and
micro-fluidics, to name a few. In all such applications, detailed knowledge regarding the stability of the constrained capillary interface is essential. As could be
expected, the extent and type of wetting-contact can significantly influence both
the stability and dynamics.
1.1
Motivation
One typically encounters capillary interfaces in courses on hydrodynamic stability when being introduced to classic results, such as the Rayleigh oscillations of
a spherical drop (Rayleigh, 1879) or Plateau-Rayleigh break-up of a liquid cylinder (Plateau, 1863; Rayleigh, 1879). These classic examples will be referred to
as ‘unconstrained’ problems, because the respective interface does not contact a
solid-support or the fluid domain is isolated. In contrast, a ‘constrained’ interface
partially contacts a solid-support and also has a three-phase contact-line, defined
as the geometric curve formed at the intersection of the solid, liquid and gas phases.
With regards to terrestrial applications, most technologies that employ capillary
interfaces are constrained and the classical results (unconstrained) can be of lim1
ited utility, but may certainly be used as a guide to understanding the physics.
In general, the constrained problem is more difficult to analyze than the unconstrained problem, because one must establish the appropriate governing equations
at the contact-line. Two such models are used in this thesis. The first assumes the
contact-line is immobile or ‘pinned’, while the second has a dynamic contact-line,
whose motion is governed by a prescribed constitutive law.
The problems considered in this thesis are motivated by a number of scientific
phenomenon and technological innovations, which can be related to the classic
(unconstrained) problems mentioned above. Some applications which utilize drops
include; adaptive liquid lens focusing for optical devices (Lopez & Hirsa, 2008),
droplet ejection for ink-jet printing (James et al., 2003b) and drop atomization
for spray-cooling and fuel injection (Vukasinovic et al., 2007). Similarly, both
the crown-splash problem (Deegan et al., 2008) and the upstream meniscus in
the planar-flow melt-spinning device (Steen & Karcher, 1997) can be viewed as
a constrained cylindrical-like interface. The aforementioned applications, which
all have immobile or pinned contact lines, require a thorough understanding of
the dynamics of the constrained capillary interface to ensure maximal control over
the respective process. For example, the adaptive liquid lens consists of a system
of coupled spherical-cap droplets, whose focal length can be quickly and continuously varied by acoustic (pressure) pulses (Lopez et al., 2005). To efficiently
process images from this device, the frequency spectrum and corresponding interface deformations are required. This information, which is unavailable from the
classic (unconstrained) problem (Rayleigh, 1879), is computed here.
In addition to numerous industrial applications, the moving contact-line is also
of great scientific interest, because of the limited understanding of the physics there.
2
One of the most commonly accepted constitutive laws relates the contact-angle to
the contact-line speed. In this thesis, application of this particular constitutive
law is shown to produce a number of distinct results, which may be used to validate or invalidate said law. With regards to applications, the dynamics of the
moving contact-line are important for droplet manipulation in microfluidics and
coating processes. For example, a ratchet-like translational motion (instability)
is observed for sessile drops on chemically-treated gradient surfaces (Daniel et al.,
2004). Similarly, Noblin et al. (2009) have shown that this translational motion can
be directionally controlled by both horizontal and vertical forcing, while Brunet
et al. (2009) demonstrated that under external forcing a sessile drop can overcome
the influence of gravity and be driven up a sloped incline against gravity. The
translational droplet motion mentioned above may be related to a ‘walking’ instability for sessile drops that is catalogued in this thesis and has not been previously
reported. Finally, rivulets of liquids on solid surfaces, whose stability is important
to heat transfer applications, can be viewed as constrained cylinder-like interfaces
with dynamic contact-lines (Davis, 1980; Weiland & Davis, 1981).
Lastly, the dynamic stability calculations for constrained capillary interfaces
under both contact-line models can become quite involved and are generally handled using computational-based approaches, such as finite element methods. The
solution methods developed in this thesis are straightforward and essentially analytic, which make the computations tractable. For example, by formulating the
constrained problem in the unifying framework of functional analysis one may
smoothly interchange between contact-line models without significantly changing
the solution method. Finally, a bounding technique is developed to generate a sufficient condition for the static stability of the catenoid without explicitly calculating
the second variation.
3
1.2
Mathematical background
When referring to the stability of an equilibrium configuration one needs to distinguish between static or dynamic stability. While the two types of stability may
be interpreted differently there certainly exists situations where the linear stability
results are coincident, as is the case for an interface of uniform surface tension σ
holding an underlying fluid.
To establish this parallel, consider static stability first. As postulated by Young
and Laplace, a capillary surface x = r (u, v) defined with surface coordinates u, v
behaves like an elastic membrane with an energy U proportional to its surface area
Z
1
U = |r u × r v |.
(1.1)
σ
Γ
The system is in equilibrium if the first variation of surface energy
Z
Z
1
y
δU = (κ1 + κ2 − µ) y + (n · n1 − cos α)
,
σ
sin α
Γ
γ
(1.2)
vanishes for all allowable disturbances y. Here the constant µ is a Lagrange multiplier, necessary to ensure volume conservation and related to the pressure on the
interface Γ by the Young-Laplace equation
p/σ = κ1 + κ2 ≡ 2H,
(1.3)
which relates the pressure to the sum of the principal curvatures κ1 , κ2 of the
interface. Equivalently, surfaces of constant mean curvature H are necessarily in
static equilibrium. Similarly, a balance of forces on the contact line γ yields the
Young-Dupré equation
n · n1 =
σsg − σsl
≡ cos α,
σlg
(1.4)
which relates the surface normals n, n1 and surface tensions of the solid-liquid σsl ,
solid-gas σsg and liquid-gas σlg ≡ σ interfaces to the contact-angle α, as shown in
4
(a)
(b)
Figure 1.1: Three-phase contact-line: (a) definition sketch and (b) force balance on contact-line.
figure 1.1(b). The contact-angle is a macroscopic quantity specified by the material
properties of the three-phase contact-line.
The disturbance energy associated with a deviation y (δx in figure 1.2) from
the equilibrium surface configuration is proportional to the second variation
1 2
δ U=
σ
Z
Γ
£ ¡ 2
¢
¤
− κ1 + κ22 y − ∆Γ y y +
Z µ
γ
¶
∂y
+ χy y,
∂e
(1.5)
which is conveniently decomposed into its respective interface Γ and contact-line
γ contributions (Myshkis et al., 1987). Here ∆Γ is the Laplace-Beltrami operator,
1 ∂
∆Γ y ≡ √
g ∂uα
µ
√
gg
αβ
∂y
∂uβ
¶
,
(1.6)
gαβ ≡ xα · xβ is the surface metric, and χ is related to the contact-angle α and
normal curvatures (defined in Kreyszig, 1991) of the interface k and surface-of-
5
support k̄,
χ≡
k cos α − k̄
.
sin α
(1.7)
Stability of the equilibrium surface is determined by the sign of the second variation
(1.5) and can be greatly influenced by the disturbance class to which the interface
is subject. In particular the two types of contact-line disturbance that have no
associated disturbance energy are of particular interest(c.f. figure 1.2). The first
satisfies
∂y
+ χy = 0 [γ],
∂e
(1.8)
and preserves the static contact-angle α in accordance with the Young-Dupré equation (1.4) and is termed the ‘natural disturbance’ (c.f. figure 1.2(a)). The natural
contact-line condition (1.8) represents the linearization of the Young-Dupré equation (1.4) about the angle α. Equation (1.8) is derived in Appendix A. The second
has a ‘pinned’ contact-line
y = 0 [γ],
(1.9)
as shown in figure 1.2(b). When the equilibrium surface is subject to either of the
aforementioned disturbances, stability is uniquely determined by the sign of the
eigenvalue λ of the boundary value problem
¡
¢
−∆Γ y − κ21 + κ22 y = λy,
(1.10)
with stability assured if λ > 0.
With regards to dynamic stability, one method frequently used to study small
interface disturbances of a fluid subject to capillarity, is to manipulate the hydrodynamic equations into an eigenvalue problem via a normal mode expansion (eλt ).
Following the analysis of Davis (1980), a disturbance energy balance will be derived
by manipulating the linearized hydrodynamic field equations. To begin, consider
an incompressible, viscous fluid occupying an arbitrary domain D, bounded by a
6
(a)
e
(b)
e1
1
"
!
x
,-
n
+!
#
(k)
$% & '()
y
*
n1
1
#"
( )
$% . '/0 )
Figure 1.2: Disturbance classes: (a) fixed contact-angle (natural) and (b)
fixed contact-line (pinned).
surface-of-support ∂Ds and an interface ∂Df (Γ) of constant surface tension, as in
figure 1.3. Given a velocity field v, interface disturbance y and growth rate λ, the
time-independent, dimensionless field equations are written as
λv = ∇ · T [D],
(1.11a)
∇ · v = 0 [D],
(1.11b)
v · n = λy [∂Df ],
(1.11c)
T · n = (2H)² n [∂Df ],
(1.11d)
v = 0 [∂Ds ].
(1.11e)
Equations (1.11a) and (1.11b) represent the linearized Navier-Stokes and continuity
equations, respectively. A kinematic condition (1.11c) relates the normal velocity
of the fluid to the interface disturbance there. Similarly, the balance of normal
7
Figure 1.3: Schematic of an arbitrary fluid domain bound by a free surface
∂Df and surface-of-support ∂Ds .
stresses at the free surface is represented by the Young-Laplace equation (1.11d).
Here the pressure p from (1.3) is replaced in (1.11d) with the full stress tensor T,
T = −p1 + µ̂D,
(1.12a)
D ≡ ∇v + (∇v)t ,
(1.12b)
with µ̂ a non-dimensional viscosity parameter. Lastly, a no-slip condition (1.11e)
is necessitated at the support surface.
An ‘energy-like’ balance equation is derived by taking the dot product of the
linearized Navier-Stokes equations (1.11a) with v,
Z
Z
Z
2
λ
|v| =
D
v·T·n−
∂D
∇v : T.
(1.13)
D
The Helmholtz decomposition theorem allows one to decompose the velocity field
v = B + ∇φ into a rotational B and irrotational ∇φ part (Joseph, 2006). Using
this decomposition, one can manipulate the disturbance energy balance into
Z
¡
∂D f
¢
λ2 φ + λµ̂n · D · n [1 + (B)n / (v)n ] − (2H)² y = 0,
8
(1.14)
which is recognized as a functional eigenvalue problem on the interface disturbance
y and takes the form of a damped harmonic oscillator
λ2 M + λΦ + K = 0,
(1.15)
with
·
Z
Z
φ y, Φ ≡ µ̂
M≡
∂Df
∂Df
(B)n
n·D·n 1+
(v)n
¸
Z
y, K ≡ −
∂D f
(2H)² y. (1.16)
The dissipation is strictly positive Φ > 0 and seen to be stabilizing when the
discriminant of (1.15) is examined. Although the dissipation Φ of the disturbance
energy balance (1.14) is amenable to a variety of approximations, the irrotational
approximation v = ∇φ of the inviscid limit (µ̂ = 0) is of particular interest. In
this limit, integral (1.14) reduces to the governing integro-differential equation,
µ ¶
µ ¶
¡ 2
¢ ∂φ
∂φ
2
−∆Γ
− κ1 + κ2
= −λ2 φ,
(1.17)
∂n
∂n
which is augmented with a boundary condition on the contact line γ. Dynamic
stability is determined from the sign of λ2 , with interface oscillations for λ2 < 0 and
instability when λ2 > 0. One should note that the structure of the static (1.10) and
dynamic (1.17) boundary value problems are identical, but the dynamic problem
is supplemented by the no-penetration condition (1.11e) on the surface-of-support.
As such, static stability can be recovered from the dynamic formulation by setting
the growth rate to zero.
In most studies, solution of the governing hydrodynamic equations (1.11) relies
heavily on computationally-intensive approaches for all but the simplest geometries. One of the primary goals of this work is to re-formulate the disturbance
energy balance (1.17) within a functional analysis framework, where the extension
to constrained interfaces is straightforward and the computations are tractable.
This formalism is used as an apparatus to study a number of related problems.
9
1.3
Organization of thesis
The content of this dissertation is conveniently organized into two types of problems, all related to constrained interfaces. Chapters 2-4 focus on immobile or
‘pinned’ contact-lines and the extent-of-wetting contact, while Chapters 5-7 are
concerned with the effect of volume and wetting conditions on the three-phase
contact-line.
Chapter 2 considers the axisymmetric oscillations of an inviscid spherical drop
pinned on a latitudinal circle-of-contact. The effect of pinning the drop is to shift
the frequencies of the unconstrained drop (Lord Rayleigh, 1879) and modify the
mode shapes according to the location of the circle-of-contact, as well as introduce a new low-frequency eigenmode. Here the center-of-mass motion, important
in application, is partitioned amongst all the eigenmodes but the low-frequency
mode is its principal carrier. A large portion of the content of Chapter 2 appears
in an article by Bostwick & Steen (2009). Chapter 3 extends the work of the
previous chapter in analyzing the oscillations of a viscous liquid drop immersed in
an immiscible fluid and constrained by an axisymmetric spherical-belt. Here the
interface is the union of a surface-of-support and two independent free surfaces,
which are coupled by the incompressibility condition and allowed to ‘communicate’ or exchange volume under the constraint. To address the transition from
free to support-surface along the drop interface, a modified set of shear boundary
conditions, which make use of an indicator function, are proposed. As the size
of the spherical-belt constraint increases from a pinned circle-of-contact, an ambiguity in a standard definition of mode number is observed at a critical belt-size,
which is interpreted from both a mathematical and physical perspective by using
a perturbed volume embedding.
10
In Chapter 4 the stability of constrained cylindrical interfaces, partially supported by a cylindrical cup-like solid, is considered. This liquid cylinder is subject
to dynamic capillary instability, including Plateau-Rayleigh break-up. The influence of the extent-of-constraint on the dispersion relation and modal structures is
reported. In the extreme, the support reduces to a wire, aligned axially, and just
touching the interface. From prior work (Davis, 1980), this constraint is known
to stabilize the Plateau-Rayleigh limit by some 13%, whereas the wave-number
of maximum growth and characteristic break-up time are estimated here. The
constraint is then bent in-plane to add a weak secondary curvature to the now
nearly-cylindrical base-state. This is referred to as the torus-lift of the cylinder.
The static stability of these toroidal equilibria, calculated using a perturbation
approach, shows that the position of constraint is crucial - constraint can stabilize
(outside) or destabilize (inside). The combined influence of secondary curvature
and wire constraint on the Plateau-Rayleigh limit is tracked. Finally, attention
is restricted to constraints that yield a lens-like cylindrical meniscus. For these
lenses, the torus-lift is used as apparatus along with a symmetrization procedure
to prove a large-amplitude static stability result. The content of Chapter 4 has
been largely disseminated in an article by Bostwick & Steen (2010).
Chapter 5 is focused on the stability of the sessile drop under a number of
contact-line conditions, including a moving contact-line modeled by a continuous
contact-angle against speed relationship. The problem is parameterized by azimuthal wave-number l, volume via the static contact-angle α (wetting) and the
boundary conditions on the three-phase contact-line, which can be controlled by
a spreading parameter Λ. Although the majority of motions are oscillatory, an
instability is found that is related to ‘walking’ or translational motion of the drop.
Cataloging this instability may be of great importance to a number of coating
11
processes. As a limiting case, a hemispherical drop given a fixed contact-angle
disturbance has characteristic oscillation frequencies, which are degenerate with
respect to azimuthal wave-number much like the Bohr model of the atom is degenerate with respect to angular momentum quantum number. This degeneracy can
be attributed to the configurational symmetry of the hemispherical base-state and
is broken by smoothly varying either i) the volume or ii) the spreading condition.
Application of the former allows one to make an analogy between the spectrum
of these ‘broken’ states and the filling order of the periodic table by energy levels, while simultaneously utilizing both degeneracy breaking mechanisms may help
to explain mode crossing behavior related to a contact-line instability observed
experimentally (Vukasinovic et al., 2007).
Chapter 6 is concerned with the instabilities of the static rivulet and their
classification. More specifically, allowable interface disturbances are decomposed
into the varicose and sinuous mode types, which are classified by their symmetry
or anti-symmetry about the vertical mid-plane, respectively. As regards stability,
the varicose mode is the more unstable mode type and has dispersion relations that
exhibit behavior typical of capillary instability or Plateau-Rayleigh break-up. That
is, there exists a fastest growing shape that is distinguished by a non-trivial axial
wave-number km 6= 0. Alternatively, the sinuous mode that preserves the static
contact-angle exhibits a different type of instability, which is not characteristic of
Plateau-Rayleigh break-up. The instability mode shape is characterized by km = 0,
has no axial dependence and may be relevant to rivulet meandering.
Lastly, the static stability of the catenoid, under a number of contact-line
conditions, is computed using a ‘bounding’ technique in Chapter 7. In lieu of
computing the second variation explicitly, elementary results from the calculus
12
of variations are used on a critical functional, obtained from the extreme-value
theorem, to generate a sufficient condition for stability using only the geometric
properties of the equilibrium shape. This approximate method recovers well-known
results to a great degree of accuracy and is used to generate new ones regarding
the fixed contact-angle disturbance class, which is destabilizing when compared
to a fixed contact-line disturbance. In fact, the entire family of catenoids are
unstable to constant pressure, fixed contact-angle disturbances. This method also
delivers stability bounds for the general axisymmetric liquid bridge with pinned
contact-lines. Although the stability bounds obtained are only approximations, the
results presented here are almost surprisingly accurate when compared to existing
literature, while the amount of work required to obtain them has been greatly
reduced when compared to traditional stability calculations.
Each chapter is intended to be self-contained and concludes with a succinct
summation of the results from that chapter.
13
CHAPTER 2
OSCILLATIONS OF A PINNED LIQUID DROP
The content of this chapter has been largely disseminated in Bostwick & Steen
(Phys. Fluids. (2009), vol. 21, p. 032108)
2.1
Introduction
It is well-known that a plucked liquid drop will oscillate, reflecting a competition
between inertia and surface tension (capillary action). The study of small, inviscid,
free oscillations of an isolated, spherical drop is attributed to Lord Rayleigh (1879).
The Rayleigh frequencies of a drop immersed in a second fluid are given by
ωn2 =
n(n − 1)(n + 1)(n + 2) σ
(n + 1)ρi + nρe
R3
(2.1)
where σ, ρi , ρe and R are surface tension, fluid drop density (interior), density of
fluid of immersion (exterior) and the radius of the un-deformed drop, respectively.
The Rayleigh mode shapes are given by the Legendre polynomials (see Lamb,
1932). The n = 0 and n = 1 modes are zero frequency modes which can be
attributed to conservation of volume and translational invariance, respectively. In
practice, the smallest non-zero frequency mode is important, because this mode
is typically the first to be excited. The lowest non-zero mode corresponds to
n = 2 for Rayleigh oscillations. Interested in low-gravity applications, Trinh &
Wang (1982) have experimentally verified (2.1). These results, which are valid
for small perturbations, have been extended to moderate amplitude oscillations
by Tsamopoulos & Brown (1983). They use domain constrained perturbations
and a Poincaré-Lindstedt expansion to report corrections to both mode shape and
frequency at second order in amplitude for both droplets and bubbles.
14
More recently, attention has been paid to spherical fluid drops under a variety
of constraints, because of applications such as inkjet printing, crystal growth and
light focusing using liquid lenses (Kuiper & Hendriks, 2004; Lopez et al., 2005;
Lopez & Hirsa, 2008). Strani & Sabetta (1984, 1988) have considered the linear
oscillations of a drop in partial contact with a spherical bowl for both the inviscid
and viscous cases. The unperturbed shape is a single spherical drop resting on
a spherical support, similar to the contact a golf ball makes with a tee. They
find that an additional low frequency vibrational mode was present (n = 1), in
comparison with isolated drops where it is zero (2.1). This low frequency mode
has been attributed to oscillatory motion of the center-of-mass which arises due
to breaking of the translational invariance of the drop. More recent works report
observations of center-of-mass motion of constrained drops. For example, Basaran
& DePaoli (1994) report oscillatory center-of-mass motion of pendant drops, while
Bian et al. (2003) also report a low frequency slosh mode for a fluid constrained to
a cylindrical tube. In addition, Courty et al. (2006) show that the ‘translational’
mode is important for contact times of bouncing spherical droplets. One should
note that in all the aforementioned studies, the free surface is a simply-connected
domain.
In contrast, Theisen et al. (2007) consider a liquid over-filling a small cylindrical hole, drilled in a flat plate, in such a way that a droplet protrudes on
either side of the plate. The top droplet is subjected to air-pressure disturbances
from a loudspeaker setting the droplet-droplet system into motion. Theisen et al.
model the system as spherical-cap drops coupled through a length L of liquid and
predict the center-of-mass motion. They compare the predicted frequencies for
small-amplitude motions against experiment, where the agreement is reasonable.
However, non-spherical shapes are common for initial large-amplitude deformations
15
and the model breaks down. The model developed here predicts such higher modes
and may be expected to be relevant to the experiments in the limit L → 0 (see
Results). The droplet-droplet configuration is important to various applications,
such as grab-and-release applications, where liquid is pumped into the small droplet
to form a liquid bridge against a substrate (grab) and then withdrawn from the
droplet to break from the substrate (release) (Vogel et al., 2005). Additionally, in
spherical-cap liquid lens applications, the focal length (radius-of-curvature) of the
liquid lens can be quickly and continuously varied for use in an optical micro-lens
device (Kuiper & Hendriks, 2004; Lopez et al., 2005; Zhang et al., 2003, 2004).
The natural axisymmetric oscillations of an inviscid liquid drop, constrained
by a latitudinal circle-of-contact, are considered here. This work is intended to
be an extension of the inviscid study of Strani & Sabetta (1984), where the key
difference is that the domain of the free surface has two components rather than a
single component. An integro-differential boundary value problem, which governs
the linearized motion of the free surface, is derived. The governing equation is then
re-formulated as a functional eigenvalue equation on linear operators, which could
be used to re-produce the results of Rayleigh and Strani & Sabetta. The respective
problems differ only by the boundary conditions which must be enforced, or the
type of ‘allowable’ disturbance to which the interface is subject.
In the case of an isolated fluid drop (Rayleigh), the relevant boundary conditions on the free surface deformation require the perturbations to be physical, or
bounded, at the north and south poles (θ = 0, π). For the Strani & Sabetta problem, the interface deformation must vanish on the bowl-of-contact, in addition to
being bounded at the north pole (θ = 0). In both cases, the resulting problems are
well-posed two-point boundary value problems. However, pinning the fluid drop
16
on a circle-of-contact gives rise to an additional boundary condition on the free
surface perturbation. This condition, in addition to requiring disturbances to be
bounded at the poles, defines a three point boundary-value problem, thus making
the problem over-determined and unlike the standard two point boundary value
problem.
A spectral method is used to compute the oscillation frequencies/mode shapes
of the functional eigenvalue problem. The additional boundary condition at the
pinned circle-of-contact is incorporated into the solution via a constrained function
space. Judicious choice of linear combinations of Legendre polynomials allows one
to generate basis functions which obey all boundary and integral conditions. and
equivalently span the constrained function space. The Legendre polynomials, or
eigenfunctions that correspond to eigenfrequencies (2.1), form a complete set and
are chosen because of the spherical symmetry of the problem. A series solution
is constructed from this constrained function space and then used to reduce the
functional eigenvalue equation to standard matrix form, using the standard L2
inner product.
The pinned circle-of-contact constraint introduces the low frequency centerof-mass mode, as well as modifies the higher frequency modes, according to the
position of the constraint. In general, the higher-order mode shapes have an associated center-of-mass motion. That is, the center-of-mass motion partitions predominantly to but not solely to the lowest-frequency mode. Higher mode shapes
carry part of the center-of-mass motion and the fraction carried depends on the
position of the constraint.
17
ΗHΘ,tL
Ζ
Θ0
R
Figure 2.1: Definition sketch for the pinned liquid drop.
2.2
Mathematical formulation
The static equilibrium shape of the droplet is a sphere of radius R, which is pinned
on a latitudinal circle-of-contact specified by θ0 , as shown in the definition sketch
(figure 2.1). The domain is the combined region internal to and external to the
droplet, D ≡ Di ∪ De ,
Di ≡ {(r, θ) |0 ≤ r ≤ R, 0 ≤ θ ≤ π} ,
(2.2a)
De ≡ {(r, θ) |R ≤ r ≤ ∞, 0 ≤ θ ≤ π} ,
(2.2b)
where no domain perturbation is needed for linear problems. The sub-domains are
separated by an interface ∂D ≡ ∂Df ∪ ∂Ds , defined as the union of a free surface
and a surface-of-support,
∂Df ≡ {(r, θ) |r = R, θ 6= θ0 } ,
(2.3a)
∂Ds ≡ {(r, θ) |r = R, θ = θ0 } .
(2.3b)
18
The interface is given a small, time-dependent, axisymmetric disturbance of
the form η(θ, t). The inner and outer fluids are inviscid and incompressible, while
the flow is assumed to be irrotational and the effect of gravity is neglected.
2.2.1
Dynamic equations
The flow is assumed to be irrotational, therefore the velocity field may be written
as v = −∇Ψ. Additionally, the assumption of fluid incompressibility dictates the
velocity potential Ψ must satisfy Laplace’s equation on the domain, written in a
spherical coordinate system as
∂
∇ Ψ = sin θ
∂r
2
µ
r
2 ∂Ψ
∂r
¶
∂
+
∂θ
µ
∂Ψ
sin θ
∂θ
¶
= 0 [D].
(2.4)
The pressure field is defined by the linearized Bernoulli equation,
p = p0 + ρ
∂Ψ
[D],
∂t
(2.5)
valid in both sub-domains. Henceforth, to distinguish between sub-domains, superscripts will be used for field quantities and subscripts for material properties.
Accordingly, ρi and ρe will represent the density of the internal and external fluids,
respectively, while pi and pe are the internal and external pressures, and so forth.
The linearized kinematic condition on the free surface relates the radial velocity
to the surface deflection there,
∂η
∂Ψ
=−
[∂Df ].
∂r
∂t
(2.6)
Similarly, the no-penetration condition on the pinned circle-of-contact requires the
radial velocity amplitude to vanish,
∂Ψ
∂η
=−
= 0 [∂Ds ].
∂r
∂t
19
(2.7)
A perturbed interface, held by uniform surface tension σ, generates pressure
gradients, which is described by the normal stress boundary condition
pi − pe = σ∇ · n̂,
(2.8)
or Young-Laplace equation for inviscid fluids (Drazin, 1981). Equivalently, the
pressure difference across the interface is balanced by the surface tension times
the mean curvature of the disturbed interface. In the limit of small deflection,
|η|/R << 1, the mean curvature evaluates to the term in the square brackets,
·
2
1
p −p =σ
− 2
R R
i
e
µ
1 ∂
sin θ ∂θ
µ
¶
¶¸
∂η
sin θ
+ 2η ,
∂θ
(2.9)
which holds on the free part of the interface ∂Df .
As the fluid is assumed to be incompressible, the interface perturbation is
constrained by a conservation of volume condition,
Zπ
η (θ) sin θ dθ = 0.
(2.10)
0
2.3
Reduced system
The mathematical model is reduced in the standard way by the use of normal
modes,
Ψ(r, θ, t) = φ(r, θ)eiωt ,
(2.11a)
η(θ, t) = iy(θ)eiωt ,
(2.11b)
where the surface disturbance is taken π/2 out of phase with respect to the radial component of velocity (2.11b), consistent with the kinematic condition (2.6).
Normal modes (2.11) are substituted into the governing hydrodynamic equations
20
(2.4)-(2.10) to generate a reduced problem on the new functions φ and y;
∇2 φ = 0 [D] ,
·
µ
¶
¸
¡ i
£ f¤
¢
σ
1 ∂
∂y
e
ω ρi φ − ρe φ = − 2
sin θ
+ 2y
∂D ,
R sin θ ∂θ
∂θ
£
¤
dφ
= ωy ∂Df ,
dr
dφ
= ωy = 0 [∂Ds ] ,
dr
Zπ
y (θ) sin θ dθ = 0,
(2.12a)
(2.12b)
(2.12c)
(2.12d)
(2.12e)
0
which is recognized as an eigenvalue problem on allowable interface perturbations.
2.3.1
Velocity potential solution
The reduced velocity potential φ obeys Laplace’s equation on the domain and
(2.12a,2.12c) is recognized as a standard Neumann type boundary value problem.
Introducing the coordinate transformation µ = cos θ, separation of variables and
the method of Frobenius may be used to find a standard solution,
Ã
!
∞
k
X
ξ
r
k
φi (r, µ) = ωR ξ0 +
Pk (µ) ,
k Rk
k=1
̰
!
X ξk Rk+1
φe (r, µ) = −ωR
Pk (µ) ,
k + 1 rk+1
k=1
(2.13a)
(2.13b)
where
ξk =
(y, Pk )
2
, (Pk , Pk ) =
.
(Pk , Pk )
2k + 1
Here (f, g) is the standard inner product of square integrable functions on the
domain µ ∈ (−1, 1),
Z
1
(f, g) =
f (µ) g (µ) dµ.
(2.14)
−1
One should note that (2.13) is a general solution for an arbitrary surface perturbation.
21
2.3.2
Surface deformation
The balance of capillary and inertial pressures is represented by (2.12b) and is
expanded using the new coordinate µ to give
µ
¶
·
¸
¢
d ¡
ρi ωR2 i
ρe e
2 dy
1−µ
+ 2y = −
φ (R, µ) − φ (R, µ) .
dµ
dµ
σ
ρi
(2.15)
The velocity potential solution (2.13) is used to reduce the inhomogeneous differential equation (2.15) to an integro-differential equation
"∞ µ
#
2
X 1 ρe 1 ¶ (y, Pk )
dy
d
y
+ 2y = −λ2
+
Pk ,
(1 − µ2 ) 2 − 2µ
dµ
dµ
k
ρ
(P
i k+1
k , Pk )
k=1
(2.16)
governing the free surface deformation, where λ2 ≡ ρi ω 2 R3 /σ is the scaled eigenfrequency.
Boundary/Integral conditions
Allowable solutions of (2.16) necessarily satisfy the following boundary conditions.
All solutions must be bounded at the north and south poles
y(±1) − bounded.
(2.17)
The no-penetration condition (2.12d) on the pinned circle-of-contact reduces to a
zero-amplitude perturbation
y(ζ) = 0,
(2.18)
with ζ ≡ cos θ0 . Additionally, the conservation of volume constraint (2.12e) reduces
to
Z
1
y (µ) dµ = 0.
(2.19)
−1
Equations (2.16)-(2.19) represent the reduced eigenvalue problem on the free surface perturbation.
22
2.4
Solution method
To compute the characteristic oscillation frequencies and corresponding mode
shapes, it is productive to frame the integro-differential equation as an operator
equation,
K [y] + λ2 M [y] = 0,
(2.20)
with the linear operators defined via (2.16) as
d2
d
K [y] ≡ (1 − µ2 ) 2 y − 2µ y + 2y,
dµ
dµ
·
¸ X
¶
∞ µ
ρe
1 ρe 1
(y, Pk )
≡
M y;
+
Pk .
ρi
k
ρ
(P
i k+1
k , Pk )
k=1
(2.21a)
(2.21b)
The operators K and M are self-adjoint and M is a positive operator, while ρe /ρi
is the density ratio and may be treated as a parameter. If the functions y are
restricted to boundary conditions (2.17) and the volume constraint (2.19), then
the eigenvalue problem corresponds to the classical Rayleigh problem. Solution
gives Legendre polynomials as eigenfunctions, which are the Rayleigh modes, and
eigenvalues which correspond to the Rayleigh frequencies (2.1).
Alternatively, the problem may be posed as a variational one following a standard Rayleigh-Ritz procedure (Segel, 1987). The required input to the Ritz method
is a set of functions, which span a predetermined function space. The strategy employed will be to construct a constrained function space that incorporates the
additional boundary condition (2.18) on the pinned circle-of-contact. Notice that
this restriction does not affect the velocity potential solution (2.13), since this a
general solution to the Neumann boundary value problem for an arbitrary surface disturbance. Finally, a series solution to (2.20) is sought using appropriately
chosen basis functions that span the constrained function space.
23
2.4.1
Constrained function space
A set of functions which span the constrained function space is derived here. The
idea is to construct basis functions hk (µ) satisfying (2.17), (2.18) and (2.19). Then
a solution of the following form is sought,
X
y (µ) =
fk hk (µ).
(2.22)
k=1
The procedure is straightforward but some details of the steps involved are offered.
To construct the function space, begin by assuming a test function of the form
g(µ) =
N
X
ci Pi (µ).
(2.23)
i=0
The Legendre polynomials Pi (µ) are used because they form a complete, orthonormal set on the domain and they identically satisfy the boundedness condition
(2.17).
Consider the conservation of volume condition (2.19) first. Substitution of the
test function g(µ) into (2.19) gives
Z
Z
1
Since
−1
R1
−1
Z
1
1
PN (µ)dµ = 0. (2.24)
P1 (µ)dµ + · · · + cN
P0 (µ)dµ + c1
g(µ)dµ = c0
−1
Z
1
−1
−1
Pi (µ)dµ = 0 for all i 6= 0 (MacRobert, 1967), the only contribution from
(2.24) is the first term and c0 = 0 is determined.
Next, enforcing the no-penetration condition (2.18) requires
g(ζ) = c1 P1 (ζ) + c2 P2 (ζ) + · · · + cN PN (ζ) = 0.
(2.25)
This algebraic equation may be interpreted as the inner product between a fixed
vector and the unknown coefficient vector [c1 , c2 , ...cN ]. Equivalently, it says there
are N − 1 linearly independent coefficient N − 1 dimensional vectors that solve
24
(2.25). In other words, there are N − 1 basis functions, parameterized by the
location of the pinned circle-of-contact ζ, and conveniently written as
vk (µ) = Pk (µ) −
Pk (ζ)
P1 (µ), k = 2, 3, · · · , N.
P1 (ζ)
(2.26)
It is readily verified that these functions satisfy (2.17), (2.18), and (2.19). As
stated above, N − 1 such functions span the reduced space (space with constraint
incorporated) or, alternatively, if one function in the direction of the constraint is
appended, these N functions will span the original unconstrained space.
The basis functions (2.26) are linearly independent but they are not orthonormal. For efficiency in computations, it is convenient to work with an orthonormal
set. The last step is to use a Gram-Schmidt procedure on the set of functions vk
to deliver orthonormal functions, hk , renumbered k = 1, 2, · · · N − 1. This step is
done using symbolic computer algebra.
2.4.2
Reduction to matrix form
The variational approach of Rayleigh-Ritz generates a set of algebraic equations
from the truncated solution series expansion,
y (µ) =
N
−1
X
fk hk (µ).
(2.27)
k=1
Note that this is an order N −1 approximation in the constrained space but an order
N approximation in the unconstrained space. That is, the surface perturbation
has N − 1 degrees-of-freedom, all orthogonal to the constraint direction (the N th
dimension).
The operator equation (2.20) is reduced to a matrix equation
Kij fj = −λ2 Mij fj ,
25
(2.28)
by taking inner products with the basis functions,
Z
Z
1
Kij ≡
1
K [hi ] hj dµ, Mij ≡
−1
M [hi ] hj dµ.
(2.29)
−1
For example, the 1j position is the projection onto the j th basis function, when
the 1st basis function is applied to the operator. Solutions of (2.28) determine
the eigenvalues λ2n and the eigenvectors fˆ(n) , from which the eigenfunctions yn are
(n)
readily constructed by applying the eigenvector coefficients fˆk to the orthonormal
basis functions hk ,
yn (µ) =
N
−1
X
(n)
fˆk hk (µ).
(2.30)
k=1
As the eigenfunctions are determined only up to a constant, the final step is to
fix that constant by specifying
ŷn =
yn
.
max (|yn (−1)|, |yn (1)|)
(2.31)
This scaling is reminiscent of the Rayleigh eigenfunctions where a similar scaling
is applied to the north pole only (µ = 1). The scaling used here is with respect to
the maximum displacement at either pole (µ = ±1) because, unlike the Legendre
polynomials (Rayleigh eigenfunctions), the eigenfunctions are neither symmetric
nor antisymmetric about the equator and because the norm on the eigenfunctions
is inherited from the norm on the Legendre polynomials.
A number of consistency checks on the computational results are performed,
as well as a comparison of limiting cases with previous results from the literature.
These will be discussed in the next section. Another check uses the self-adjoint
nature of the operator K. The null space of K is found to be P1 (µ), which has a
node at ζ = 0, thus satisfying the pinned condition (2.18). The Fredholm alternative applied to (2.20) requires that the right hand side, λ2 M [y], be orthogonal to
26
the nullspace of K; that is,
¡
where
M [hj ] ≡
¢
P1 (µ), −λ2 M [hj ] = 0
X µ1
k=1
ρe 1
+
k ρi k + 1
¶·
(2.32)
¸
(hj (µ), Pk (µ))
Pk (µ).
(Pk (µ), Pk (µ))
In accordance with the positive definite structure of M , the term in the brackets on
the right hand side is non-zero, which requires that λ2 = 0. Computational results
show that the n = 1 eigenfrequency tends to zero when the circle-of-contact tends
to the equator of the fluid drop (ζ = 0), consistent with the Fredholm alternative
requirement.
2.5
Results
The vibrational frequencies and respective mode shapes of the constrained fluid
drop are computed. The scaled frequencies λ2n for the first three modes are plotted
in figure 2.2 as a function of the location of the circle-of-contact ζ, for ρe /ρi = 0.
As could be anticipated, the frequencies are symmetric with respect to the pin
location, λ2n (ζ) = λ2n (−ζ). For reference, the corresponding Rayleigh frequencies
(R in the legend) are plotted as horizontal lines. The frequency of the constrained
problem is never lower than that of the unconstrained problem (Griffel, 1985),
with equality achieved whenever the constraint falls on a node of the corresponding Rayleigh mode (Legendre polynomial). This is to be anticipated since the
constraint is satisfied ‘naturally’ in the latter case. What is not anticipated is
that the constrained frequency can more than double the Rayleigh frequency for
certain pin locations. For example, pinning the n = 2 mode at the equator raises
the frequency from 8 to 22.
27
n‡1
n‡2
n‡3
n‡2R
n‡3R
Λ2
60
50
40
30
20
10
0
0.2
0.4
0.6
0.8
1
Ζ
Figure 2.2: Eigenfrequency against pin location (ρe /ρi = 0).
Much like the local minimums of figure 2.2 which occur at ‘natural’ pin locations, the local maximums occur at ‘unnatural’ pin locations with respect to
the ‘unconstrained’ Rayleigh modes. The definition of an ‘unnatural’ pin location
becomes apparent when the eigenfrequencies and eigenmodes are computed. The
results presented here were obtained with a 13 term expansion, which gives convergence of the first three eigenfrequencies to within 0.1% for all pin locations.
The rate of convergence depends upon the pin location. For example, the n = 2
mode shows convergence to the prescribed tolerance (0.1%) using only 5 terms for
ζ = −0.557, a ’natural’ location for this mode. On the other hand, if the drop
is pinned at the equator, the n = 2 mode converges to the prescribed tolerance
using 10 terms. If one desires the eigenfrequency and mode shape for a high wavenumber, say n = 20, the number of terms necessary in the expansion is of the
order 30. However, the recursive nature of the Gram-Schmidt procedure makes it
very computationally intensive to generate 30 orthonormal basis functions; this is
the step that computationally limits the approach.
Figure 2.2 also shows that pinning the drop introduces a low frequency mode
28
(a)
(b)
(c)
(d)
(e)
(f )
Figure 2.3: Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = −0.99.
Mode n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ).
(n = 1) not present in the unconstrained Rayleigh problem. This mode has been
reported previously by Strani & Sabetta (1984), among others. The Strani study
considered a drop in partial contact with a spherical cap support, remarking that
the new mode ‘tends to a zero-frequency rigid displacement’ as the constrained
portion of the surface vanishes. The Strani limit of the contact region shrinking
to a point (θ → π) coincides with the limit of the circle shrinking to either pole
(ζ → −1). These results agree with Strani’s in this common limit and both results
tend to the corresponding frequencies (2.1). Mode shapes and velocity potentials
for this case are shown in figure 2.3.
Another special pinning location is the equator. For the Rayleigh problem, all
odd modes have a node at µ = 0. Therefore, when ζ = 0, all odd constrained
29
(a)
(b)
(c)
(d)
(e)
(f )
Figure 2.4: Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = 0.
Mode n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ).
frequencies correspond to odd Rayleigh frequencies. The mode shapes for n =
1 and n = 3, P1 (µ) and P3 (µ), respectively, are shown in figure 2.4 above the
corresponding velocity potentials. Figure 2.5 shows the first three eigenfunctions
for ζ = −0.557, which is the location of a node for P2 (µ). Not surprisingly, the
n = 2 mode shape corresponds to P2 (µ).
2.5.1
Decomposition of eigenmodes
In view of the completeness of the Legendre polynomials, any shape can be decomposed into a weighted sum of Pn (µ). These weights, for the mode shapes shown in
figures 2.3, 2.4, and 2.5 are computed for reference. A given eigenmode ŷn (µ) may
30
(a)
(b)
(c)
(d)
(e)
(f )
Figure 2.5: Eigenmode (a, b, c) and velocity potential (d, e, f ) for ζ = −0.557.
Mode n = 1 (a, d), n = 2 (b, e) and n = 3 (c, f ).
be decomposed into the Rayleigh modes as follows,
ŷn (µ) =
∞
X
bk Pk (µ),
(2.33)
k=1
with
bk =
(ŷn , Pk )
2k + 1
=
(ŷn , Pk ) .
(Pk , Pk )
2
(2.34)
Shown below is the decomposition of the first four eigenmodes into the first 8
coefficients of the 13 term expansion, for the pin locations used to generate figures 2.3-2.5.
31
Rayleigh decomposition for ζ = −0.99
n
b1
b2
b3
b4
b5
b6
b7
b8
1
0.784
0.321
-0.160
0.108
-0.080
0.061
-0.048
0.038
2
-0.084
0.804
0.383
-0.188
0.127
-0.095
0.073
-0.057
3
-0.031
0.128
-0.783
-0.453
0.206
-0.136
0.100
-0.076
4
0.015
-0.056
0.152
-0.719
-0.511
0.212
-0.135
0.097
Rayleigh decomposition for ζ = 0.001
n
b1
b2
b3
b4
b5
b6
b7
b8
1
0.999
0.001
0
0
0
0
0
0
2
0
0.644
-0.008 0.523
0.002
-0.214
-0.001
0.132
3
0
0.004
0.991
0.006
0
-0.002
0
0.001
4
0
-0.104
0
0.589
-0.028
0.570
0.002
-0.222
Rayleigh decomposition for ζ = −0.557
2.5.2
n
b1
b2
b3
b4
b5
b6
b7
b8
1
0.488
-0.072
0.312
-0.189
0.018
0.084
-0.085
0.023
2
0.034
0.928
0.064
-0.034
0.003
0.014
-0.014
0.004
3
0.039
0.010
-0.619
-0.499
0.028
0.108
-0.101
0.026
4
-0.027 -0.006
0.177
-0.482
-0.476
-0.377
0.269
-0.062
Center-of-mass motion
Although a number of authors have reported the low frequency n = 1 mode, most
have incorrectly labeled this mode as a center-of-mass mode. Here the definition
32
of the center-of-mass is used to decompose the eigenmodes into center-of-mass
motion, from which one can calculate how this motion partitions amongst mode
shapes.
The center-of-mass resides on the z-axis, due to the assumed axisymmetry of
perturbations. Hence, one may define the center-of-mass of a given perturbation
η(θ, t) in the standard way as
Z
Z
mzcm = z dm = 2πρ
π
Z
0
r
r3 sin(θ) cos(θ) dr dθ
0
ρπ
=
2
Z
π
(2.35)
4
r sin(θ) cos(θ) dθ,
0
where the radial perturbation coordinate is defined as
r = R (1 + ²η(θ, t))
(2.36)
r4 = R4 (1 + ²η(θ, t))4 = R4 [1 + ² (4η(θ, t)) + · · · ] .
(2.37)
with
Likewise, the center-of-mass coordinate may be expanded as
zcm = (zcm )0 + ² (zcm )1 + · · · .
(2.38)
Expanding (2.35) in ² gives
Z
ρπR4 π
m [(zcm )0 + ² (zcm )1 + · · · ] =
sin(θ) cos(θ) dθ
2
0
Z π
4
+²2πρR
η(θ, t) sin(θ) cos(θ) dθ.
(2.39)
0
Using µ = cos(θ), m =
ρ 34 πR3
and equating terms of the same order we find
(zcm )0 = 0,
(2.40a)
3
(zcm )1 = R (ηn (µ, t), P1 (µ)) .
2
(2.40b)
Finally, one can use the definition of ηn (µ, t) as an eigenmode,
ηn (µ, t) = ieiωn t ŷn (µ),
33
(2.41)
to show the scaled center-of-mass motion is
(zcm )1
3
= ieiωn t (ŷn , P1 ) .
R
2
(2.42)
For the Rayleigh problem, the n = 1 mode has zero frequency and is correctly
associated with the linearized center-of-mass motion of the drop (2.1). The imposed constraint, fixed in the laboratory frame, breaks the translational invariance
to typically yield a non-zero frequency for the center-of-mass motion. As such,
the center-of-mass motion for the Rayleigh modes is completely determined by the
n = 1 mode and the higher order mode shapes are effectively decoupled for motion
of the center-of-mass. In contrast, there is no such simple partition for constrained
motions.
Of particular interest is in what role the center-of-mass plays in the higher
modes that are reported here. To that end, the contribution of the center-of-mass
motion (zcm ) to any mode is found to be, (zcm )/R=3/2 ieiωn t (ŷn , P1 ). Understanding this contribution is of utmost importance in applications such as ink-jet printing
where generating large excursions of the center-of-mass often correlates with pinchoff of droplets. Despite the fact that this analysis is restricted to small-amplitude
motions, such linear results are known to often carry through to nonlinear behavior.
The center-of-mass is found to oscillate at the eigenfrequency ωn with a
contribution to the eigenmode (ŷn , P1 ).
Figure 2.6 shows the decomposition
(zcm )n = (ŷn , P1 ) as a function of pinning location, which shows the n = 1 mode
carries the majority of the center-of-mass motion, but not all of it. The higher
mode shapes are also accompanied by an associated motion of the center-of-mass,
thereby coupling the translation and oscillatory Rayleigh modes. Courty et al.
(2006) have postulated that this coupling might explain the discrepancy between
experimental (Richard et al., 2002) and theoretical pre-factor values for contact
34
(a)
Hzcm Ln
(b)
Hzcm Ln
n‡1
n‡2
n‡3
n‡4
1
0.8
0.2
n‡2
n‡3
n‡4
0.6
0.1
0.4
0.2
0.2
0.4
0.6
0.8
1
Ζ
0.2
0.4
0.6
0.8
1
Ζ
Figure 2.6: Center-of-mass motion contribution to eigenmodes as a function
of pin location ζ.
and impact times for droplet bouncing. While it may be convenient to view the
n = 1 mode as a center-of-mass mode, it is clearly an approximation, at best.
Although the n = 1 mode carries the majority of the center-of-mass motion, the
ordering of the center-of-mass decomposition for the n > 1 modes is not preserved
with pin location (figure 2.6 (b)). For example, the n = 2 mode carries less centerof-mass motion than the n = 3 or n = 4 mode for a pin location near ζ = 0.557 (the
node of P2 (µ)). For a fixed initial deformation amplitude and/or kinetic energy,
suppose one would like to excite a preferred mode to get the greatest extension
of the center-of-mass in order to encourage a pinch-off of a certain volume, say.
Figure 2.6 provides a guide as how to choose the optimal pin location for such
behavior.
2.5.3
Extension to a double pinned fluid drop
One feature of the spectrum shown in figure 2.2 are the regions of inaccessible
frequency space, such as 22 < λ2 < 30. As has been demonstrated, the discrete
35
spectrum of the free drop may be broken through the introduction of a pinned
circle-of-contact. This observation leads one to believe these gaps in frequency
space could be filled by introducing another pinned circle-of-contact, or the entire
frequency space could be accessible.
The analysis outlined previously can be extended to the case of a double-pinned
spherical fluid drop, where the locations of the pinned circles-of-contact are defined
as ζ1 and ζ2 . To proceed, (2.17)-(2.19) are augmented with a boundary condition
on the radial velocity amplitude or equivalently the perturbation amplitude at
the second pinned circle-of-contact. Now, the boundary/integral conditions to be
satisfied are:
y(ζ1 ) = 0,
(2.43a)
y(ζ2 ) = 0,
Z 1
y (µ) dµ = 0.
(2.43b)
(2.43c)
−1
By the same reasoning as section 4, a set of basis functions are constructed
which obey all necessary conditions and then used to form an approximate solution
to the integro-differential operator equation. Again, begin by assuming a test
function of the form,
g(µ) =
N
X
ci Pi (µ).
(2.44)
i=0
As was the case for the single pinned circle-of-contact, (2.43c) requires c0 = 0.
Next, substituting (2.44) into (2.43a,2.43b) gives
g(ζ1 ) = c1 P1 (ζ1 ) + c2 P2 (ζ1 ) + · · · + cN PN (ζ1 ),
(2.45a)
g(ζ2 ) = c1 P1 (ζ2 ) + c2 P2 (ζ2 ) + · · · + cN PN (ζ2 ),
(2.45b)
36
(a)
Λ
(b)
n‡1
n‡2
n‡3
2
100
n‡1
n‡2
n‡3
80
Λ
100
80
60
60
40
40
20
20
-1 -0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
Ζ2
2
-1 -0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
Ζ2
Figure 2.7: Frequency against second pin location ζ2 , while holding the first
pin location (a) ζ1 = 0.01 and (b) ζ1 = 0.15 fixed.
or equivalently




 P1 (ζ1 ) P2 (ζ1 ) · · · PN (ζ1 )  



P1 (ζ2 ) P2 (ζ2 ) · · · PN (ζ2 ) 

c1
..
.
cN
 

  0 
 =  .


0
(2.46)
It can be shown that the solution of (2.46) gives N − 2 linearly independent
basis functions,
vn (µ) = Pn (µ) −
Pn (ζ1 )
(P1 (ζ1 )Pn (ζ2 ) − Pn (ζ1 )P1 (ζ2 ))
P1 (µ) −
P2 (µ)
P1 (ζ1 )
(P1 (ζ1 )P2 (ζ2 ) − P2 (ζ1 )P1 (ζ2 ))
P2 (ζ1 ) (P1 (ζ1 )Pn (ζ2 ) − Pn (ζ1 )P1 (ζ2 ))
+
P1 (µ)
P1 (ζ1 ) (P1 (ζ1 )P2 (ζ2 ) − P2 (ζ1 )P1 (ζ2 ))
(2.47)
which satisfy (2.43a)-(2.43c). Again the Gram-Schmidt procedure is applied to
these linearly independent functions to generate a set of orthonormal functions,
which are applied to the operator equation (2.20) to reduce the problem to a
truncated set of algebraic equations.
Shown in figures 2.7,2.8 are the computed vibrational frequencies as a function
of the second pin location ζ2 , while holding the first pin location ζ1 fixed. As postulated, the introduction of the second pinned circle-of-contact leads to the filling
37
(a)
(b)
Λ2
n‡1
n‡2
n‡3
Λ
n‡1
n‡2
n‡3
2
80
100
80
60
60
40
40
20
20
-1 -0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
Ζ2
-1 -0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
Ζ2
Figure 2.8: Frequency against second pin location ζ2 , while holding the first
pin location (a) ζ1 = −0.5 and (b) ζ1 = −0.7 fixed.
(a)
(b)
(c)
Figure 2.9: Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −.55, ζ2 = 0.55.
of the frequency space, previously inaccessible with only one circle-of-contact, as
shown in figure 2.2. Likewise, sample mode shapes are given in figures 2.9-2.11.
2.5.4
Density variation
The above results have illustrated the behavior for ρe = 0. Results for other density
ratios are readily computed. Increasing the ratio of outer to inner density (ρe /ρi )
is found to decrease the eigenfrequencies compared to the case of an isolated fluid
38
(a)
(b)
(c)
Figure 2.10: Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −0.775, ζ2 = 0.001.
(a)
(b)
(c)
Figure 2.11: Eigenmodes (a) n = 1 (b) n = 2 and (c) n = 3 with pin locations
ζ1 = −0.91, ζ2 = 0.1.
drop (ρe = 0). To illustrate, let λ2n,0 denote the nth eigenfrequency for ρe = 0 and
λ2n be the eigenfrequency for ρe 6= 0. Using the definition of λ2 , define Ω2 ≡ λ2n,0 /λ2n
as the ratio of the eigenfrequency for an isolated drop to the eigenfrequency for
ρe 6= 0 . The computed Ω2 against (ρe /ρi ) is shown in figure 2.12. It is seen that a
non-zero outer density has a greater effect on the higher frequency modes, which
is consistent with the numerical results of Strani & Sabetta for varying density
ratios in the common limit, ζ → −1 (figure 2.12).
39
(a)
(b)
W2
W2
10
n‡1
n‡2
n‡3
8
n‡1
n‡2
n‡3
8
6
6
4
4
2
2
Ρe
0
2
4
6
8
Ρe
0
10 Ρi
2
4
6
8
10 Ρi
Figure 2.12: Density variation in eigenmodes for a drop pinned on the (a)
south pole ζ = −0.99 and (b) equator ζ = 0.
2.6
Concluding remarks
The classic Rayleigh problem can be posed as a variational problem for a quadratic
functional, where the minimization is taken over functions that satisfy conditions
(2.17) and (2.19). This is referred to as the unconstrained problem. The integrodifferential equation governing the motion of the free surface has been formulated as
an operator equation (2.20), which represents the corresponding Euler-Lagrange
equation, obtained by requiring the first variation to vanish. As an extension
to the unconstrained Rayleigh problem, candidate functions have been restricted
to be pinned on a latitudinal circle-of-constraint (2.18). Implementation of the
Rayleigh-Ritz variational approach allows one to show that the frequency of the
constrained problem cannot decrease relative to the unconstrained one (Griffel,
1985). The eigenvalues and eigenfunctions of the constrained problem that have
been presented here are consistent with these bounds.
One motivation for this study is to compare against predictions of the sphericalcap model. Theisen et al. (2007) restrict to spherical-cap shapes in considering the
dynamics of the center-of-mass motion of two droplets coupled through a tube.
40
Λ2
n‡1
SC
4
3
2
1
0
0.2
0.4
0.6
0.8
1
Ζ
Figure 2.13: Frequency versus pin location: Spherical cap (SC) and n = 1
mode.
In the limit of zero tube length, the system reduces to two spherical caps coupled
along a common circle-of-contact. When the two caps are complementary pieces of
a sphere, the equilibrium state under consideration corresponds to a sphere with a
latitudinal circle-of-constraint, or the problem treated here. Among other results,
they report frequency of oscillation of small-amplitude motions as it depends on
ζ. The frequency of the computed n = 1 mode shows a qualitatively similar
dependence on ζ as the spherical-cap model (Figure 2.13). For example, both give
an oscillation frequency of zero when ζ = 0 and ζ = 1 and have a single maximum
near ζ = 0.7. However, there is quantitative discrepancy between the frequencies,
which can be traced to the different shapes allowed near the circle-of-contact. For
the spherical caps, the tangent to the interface from below and from above the
pinning latitude must be discontinuous (except for equilibrium shapes). Whereas,
the formulation developed here does not allow a singularity in curvature, which
precludes discontinuous slopes. Thus, the spherical cap shape may reasonably
capture the observed frequencies for finite non-zero tube lengths but will likely fail
quantitatively for sufficiently short tubes.
These issues are illustrated in figure 2.14. A snapshot of the two coupled drops,
41
(a)
(b)
Figure 2.14: Comparison with experiment (a) experiment and (b) n = 2
eigenmode for ζ = 0.05.
digitally modified to remove the tube, is seen in figure 2.14(a).1 Figure 2.14(b)
shows the n = 2 computed eigenmode, for a pin location deviating slightly away
from the equator to account for small gravitational effects on the static equilibrium
shape. This qualitative comparison is acceptable if one ignores the behavior near
the contact-line. The photo shows different apparent contact angles for the top
and bottom interfaces, a feature especially apparent at the right side where the top
contact-line has evidently de-pinned. Furthermore, comparison near the contactline, photo against computation, shows a different curvature. In addition, it should
be noted that, in the computed shape, the tangents at the contact-line appear
dissimilar at this scale but are actually smooth, consistent with the discussion of
the previous paragraph. Because of the contact-line issues and the influence of the
tube, quantitative comparison with the experiments is precluded. The point here
is to elucidate these issues.
1
The image from experiment in figure 2.14 (a) was taken from high-speed video recording of
oscillation experiments done in the Hirsa Laboratory at RPI (reported in Theisen et al. (2007))
42
A practical question regarding the computations presented here is whether there
is a preferred way to excite the system to get the greatest extension of the centerof-mass, say, for a fixed constraint position and initial deformation amplitude.
Clearly, the n = 1 mode carries the majority of the center-of-mass motion, but
the excursion of the n = 2, 3, · · · modes may be surprising. One might expect the
lower order modes to carry more of the center-of-mass than the higher order modes,
considering that the velocity fields for the higher order modes are more localized
on the surface (see Velocity Potentials). To the contrary, for certain pin locations,
it is seen that the higher modes carry more of the center-of-mass motion. For
these pin locations, to get the greatest extension of the center-of-mass, one should
preferentially excite particular modes.
Additionally, this analysis has been extended to include a second pinned circleof-contact. By introducing the second pinning circle, it is shown that the frequency
gaps seen in figure 2.2 begin to fill, as shown in figures 2.7,2.8. That is, according
to figure 2.2, there is no pinning location to give a scaled frequency between 22
and 30. Effectively, by adding a second constraint, one can choose a pin location
to achieve any desired frequency.
A goal of this work is to extend Strani & Sabetta’s analysis to include ‘belts’
of restricted deformation on the sphere. That is, suppose one wishes to pin the
interface between latitudes 50◦ and 70◦ . This can be done by adding a finite
number of circular pinning constraints, from which the spherical bowl constraint
considered by Strani is recovered with a sufficient number of closely placed pinning
circles.
43
CHAPTER 3
VISCOUS OSCILLATIONS OF A FLUID DROP UNDER
SPHERICAL-BELT CONSTRAINT
3.1
Introduction
Liquid drops, held by surface tension, are known to assume spherical shapes at
equilibrium. When perturbed, an isolated inviscid drop will oscillate with characteristic frequency and mode shape, given by Lord Rayleigh (1879) in the limit of
small interface deformation. The Rayleigh frequencies, which will be referred to
as the ‘unconstrained’ problem, have discrete spectrum (2.1) and corresponding
mode shapes given by the Legendre polynomials, Pn (cos θ). This theoretical result
has been verified experimentally for immiscible drops by Trinh & Wang (1982) and
free drops in microgravity by Wang et al. (1996).
Many fluids do not obey the inviscid assumption and dissipate energy, giving rise to attenuated drop oscillation amplitudes. To extend the work of Lord
Rayleigh, Lamb (1932) has computed the viscous dissipation for the free drop
under the assumption of irrotational motion and in the limit of small kinematic
viscosity ν to show the amplitude of oscillation is damped like exp(−κt), with
κ=
ν
(n − 1)(2n + 1).
R2
(3.1)
Likewise, Padrino et al. (2007) have extended Lamb’s result to report viscous corrections to the oscillation frequencies of the isolated drop in the irrotational limit.
In addition, they compare the irrotational dissipation approximation and viscous
potential flow theory to the exact solution of the linear theory obtained by Prosperetti (1980b). Both methods use a potential flow approximation, but differ in
44
how viscosity enters the governing equations. Viscous potential flow introduces
viscosity to the normal stress balance on the interface, while the irrotational dissipation approximation evaluates the viscous dissipation in the mechanical energy
equation using the irrotational potential flow solution. The irrotational dissipation
approximation is shown to be quite satisfactory for small viscosities.
Finite viscous effects, as they relate to the free drop problem, have been investigated by Reid (1960) for an isolated drop in a vacuum. In the large viscosity
limit, Chandrasekhar (1961) has established boundaries between oscillatory and
aperiodic motion. Miller & Scriven (1968) have derived a dispersion relationship
for immiscible drops, whose interface may or may not have elastic properties.
Prosperetti (1980b) has computed the spectrum of immiscible viscous drops using
a normal mode analysis. This result was later verified numerically by Basaran
(1992). In contrast to the standard normal mode analysis, Prosperetti has also
studied the initial-value problem for viscous drops and identified three-phases of
evolution: the first phase is characterized by irrotational flow, the second phase
has vorticity generated at the drop surface diffusing into the bulk and the final
phase being the least-damped normal mode (Prosperetti, 1980a,c).
The study of constrained fluid drops is predominantly motivated by applications, such as drop atomization (James et al., 2003a,b; Vukasinovic et al., 2007)
and liquid lens focusing (Lopez & Hirsa, 2008; Lopez et al., 2005), to name a
few. Most relevant to the problem considered here are the spherical constraints
used in the works of Strani & Sabetta (1984, 1988), Theisen et al. (2007) and
Bostwick & Steen (2009). Strani & Sabetta (1984) consider the linear oscillations
of a drop in partial contact with a ‘spherical-bowl’ in the inviscid limit by using
a Green’s function approach to derive an integral eigenvalue equation, which is
45
then reduced to a set of linear algebraic equations by a Legendre series expansion.
They report a new low-frequency mode, not present for isolated drops, and exponential eigenfrequency growth as the size of the spherical-bowl is increased from
a point to a fully captured sphere. Similarly, Bauer & Chiba (2004, 2005) have
also investigated spherical ‘bowl-like’ constraints for inviscid and viscous captured
drops by approximating finite-sized constraints with a large number of point-wise
constraints.
Theisen et al. (2007) study moderate amplitude spherical cap oscillations of a
drop pinned on a circle-of-contact, and report low frequency center-of-mass motions. The spherical cap model was reasonably accurate when compared against
small amplitude disturbances in their droplet-droplet experiment, but for large
initial disturbances, higher order mode shapes were shown to persist. To model
these higher order mode shapes, Bostwick & Steen (2009) analyzed the linear oscillations of a drop constrained by a latitudinal circle-of-contact and report a shift
in the characteristic frequencies compared to the unconstrained drop, as well as
the low frequency n = 1 mode. Their analysis utilizes a Rayleigh-Ritz procedure
and chosen function spaces that necessitated continuous contact angles across the
circle-of-contact constraint, whereas the droplet-droplet experiments exhibit discontinuous contact angles. This issue, among others, will be addressed here.
A drop constrained by a spherical-belt (c.f. figure 3.1) has an interface that is
decomposed into one surface-of-support and two independent free surfaces, which
may exchange volume through the underlying fluid consistent with the incompressibility condition. An integro-differential boundary value problem, governing the
free surface deformations of a drop constrained by a spherical-belt is derived and
formulated as a functional eigenvalue equation on linear operators, which takes
46
the form of a damped-harmonic oscillator. A solution to the eigenvalue problem is
found using the variational procedure of Rayleigh-Ritz on a constrained function
space, constructed to satisfy the no-penetration condition on the surface-of-support
and appropriately couple the deformations of the two free surfaces according to
volume conservation. In contrast to the unconstrained free drop, the shear boundary conditions change along the drop interface depending upon the type of surface,
either free or supported. To address this issue, a set of modified shear boundary
conditions, valid on the entire interface, are proposed and validated in the appropriate limits. While the inviscid frequencies may be computed from a standard
eigenvalue problem, the viscous frequencies are determined from a nonlinear characteristic equation, because the viscous dissipation operator is nonlinear in the
eigenvalue. The viscous frequencies are shown to bifurcate from complex to real
eigenvalues, corresponding to under-damped to over-damped motion, at a critical
value of viscosity. Higher order mode shapes are shown to bifurcate at smaller viscosities, because of strong relative motion among fluid elements, or large viscous
dissipation.
A limiting case of this analysis is the pinned circle-of-contact constraint, employed in Chapter 2. Here, in contrast, the interface may have a discontinuous
contact-angle across the pinned constraint. By comparing the computed frequencies, one learns that a pinned drop would always prefer to have a discontinuous
contact-angle, except when the circle-of-contact is placed at a ‘natural’ pin location. A ‘natural’ pin location is defined as a ‘node’ or zero of the unconstrained
eigenmode. This preference is especially pronounced at ‘unnatural’ pin locations,
observed qualitatively in the droplet-droplet experiment and an essential feature
of the spherical-cap model (Theisen et al., 2007).
47
Eigenfrequencies are readily computed as a function of spherical-belt size and
location. As the size of the constraint is increased from a pinned circle-of-contact,
the mode shapes are shown to qualitatively change their character by increasing their number of ‘nodes’, or zeroes of the corresponding eigenfunctions, while
preserving the numerical ordering of the eigenvalues at a critical belt size. In addition, the mode shape that transfers the most volume between disjoint surfaces
also changes at these points.
This chapter begins by defining the linearized hydrodynamic field equations
and relevant boundary conditions for the viscous problem, from which the equation of motion for the drop interface is derived and formulated as an eigenvalue
problem on linear operators. The functional eigenvalue equation is reduced to a
truncated set of linear algebraic equations using a Rayleigh-Ritz procedure on a
constrained function space. The eigenvalues/eigenmodes are then computed from
a characteristic equation, nonlinear in material properties and the size/location of
the constraint. The chapter concludes by offering some remarks on the computational results.
3.2
Mathematical formulation
Consider an unperturbed spherical droplet of radius R, constrained by a sphericalbelt given through the polar angle θ1 ≤ θ ≤ θ2 in spherical coordinates, as shown
in the definition sketch (c.f. figure 3.1). The drop interface is disturbed by timedependent free surface perturbations, η1 (θ, t) and η2 (θ, t), which are assumed to
be axisymmetric and small. No domain perturbation is needed for linear problems,
thus the domain is the combination of the region internal to and external to the
48
Η1 HΘ,tL
f
¶D1
Θ1
Ζ1
Θ2
Ζ2
¶Ds
R
f
¶D2
Η2 HΘ,tL
Figure 3.1: Definition sketch of a drop constrained by a spherical-belt.
static droplet;
Di ≡ {(r, θ) | 0 < r ≤ R, 0 ≤ θ ≤ π},
(3.2a)
De ≡ {(r, θ) | R < r < ∞, 0 ≤ θ ≤ π},
(3.2b)
D ≡ Di ∪ De .
(3.2c)
The interface separating the interior and exterior fluids is defined as the union of
two free surfaces and one surface-of-support;
∂D1f ≡ {(r, θ) | r = R, 0 ≤ θ ≤ θ1 },
(3.3a)
∂D2f ≡ {(r, θ) | r = R, θ2 ≤ θ ≤ π},
(3.3b)
∂Ds ≡ {(r, θ) | r = R, θ1 ≤ θ ≤ θ2 },
(3.3c)
∂D ≡ ∂D1f ∪ ∂D2f ∪ ∂Ds .
(3.3d)
The inner and exterior fluids are viscous and incompressible and the effect of
gravity is neglected.
49
3.2.1
Field equations
The field equations, governing the motion of the fluid, are written via a velocity
field u and pressure P . An incompressible fluid necessarily has a divergence-free
velocity field,
∇ · u = 0.
(3.4)
The linear momentum balance on a material volume gives the linearized NavierStokes equation
ρ
∂u
= −∇P − µ∇ × ∇ × u,
∂t
(3.5)
where the material properties, ρ and µ, are the fluid density and kinematic viscosity, respectively. Applying the curl to (3.5) gives the balance of angular momentum
ρ
∂Ω
= −µ ∇ × ∇ × Ω,
∂t
(3.6)
with the vorticity Ω defined as
Ω ≡ ∇ × u.
3.2.2
(3.7)
Velocity field definition
The velocity field for axisymmetric flows is written in spherical coordinates as
u = ur (r, θ, t) er + uθ (r, θ, t) eθ .
(3.8)
In accordance with the velocity field (3.8), the vorticity is
·
¸
∂uθ ∂ur
1
−
uθ + r
eφ .
Ω = Ω (r, θ, t) eφ =
r
∂r
∂θ
50
(3.9)
3.2.3
Reduced system
Substitution of normal modes
ur (r, θ, t) = vr (r, θ) e−γt , uθ (r, θ, t) = vθ (r, θ) e−γt ,
Ω (r, θ, t) = ω (r, θ) e−γt , P (r, θ, t) = p (r, θ) e−γt ,
η (θ, t) = y (θ) e−γt ,
(3.10)
into (3.4)-(3.7) delivers a reduced set of field equations,
∇ · v = 0,
(3.11a)
ργv = ∇p + µ∇ × ∇ × v,
(3.11b)
ργω = µ∇ × ∇ × ω,
(3.11c)
ω = ∇ × v,
(3.11d)
valid in both interior and exterior domains. Here γ is the complex growth rate.
3.2.4
Boundary/Integral conditions
The no-slip and no-penetration conditions for viscous fluids requires
vθi,e (R, θ) = 0 [∂Ds ] ,
(3.12a)
vri,e (R, θ) = 0 [∂Ds ] ,
(3.12b)
on the surface-of-support, while continuity of tangential velocity and shear stress
h
i
vθi (R, θ) = vθe (R, θ) ∂D1f , ∂D2f ,
h
i
i
e
τrθ
(R, θ) = τrθ
(R, θ) ∂D1f , ∂D2f ,
51
(3.13a)
(3.13b)
is enforced on the free surfaces. The linearized kinematic condition relates the
radial velocity to the surface deformation there
i
h
vri (R, θ) = vre (R, θ) = −γy(θ) ∂D1f , ∂D2f ,
(3.14)
and the difference in normal stress across the interface is balanced by the surface
tension σ times the linearized curvature of the surface perturbation
·
µ
¶¸
1
1
i
e
(sin θ y,θ ),θ + 2y .
τrr (R, θ) − τrr (R, θ) = −σ
R2 sin θ
(3.15)
The integral form of the incompressibility condition (3.4) constrains the interface
perturbation to be volume conserving,
Z π
y (θ) sin θ dθ = 0.
(3.16)
0
The fluids are assumed to be Newtonian, where the components of stress are related to the velocity field components. In spherical coordinates, these relationships
are
·
τrθ = τθr
³v ´ ¸
(vr ),θ
θ
=µ
+r
,
r
r ,r
τrr = −p + 2µ (vr ),r .
3.2.5
(3.17a)
(3.17b)
Velocity field decomposition
The Helmholtz decomposition theorem (e.g. Joseph, 2006) states that the velocity
field may be decomposed as the sum of rotational and irrotational fields. The
vorticity field is solenoidal and therefore may be written as the curl of a vector
potential B
ω = ∇ × B,
52
(3.18)
where for axisymmetric flows with non-trivial interface deflection (Chandrasekhar,
1961),
B = B (r, θ) er .
(3.19)
Given (3.18), the velocity field is decomposed as
v = B + ∇Ψ,
(3.20)
with the scalar field Ψ defined as the velocity potential. Let x ≡ cos(θ), then the
velocity field components are
µ
¶
µ
¶
¢
∂Ψ
1¡
2 1/2 ∂Ψ
er +
eθ .
v= B+
1−x
∂r
r
∂x
(3.21)
Similar to the field quantities, the vector and velocity potentials are expanded
with normal modes
B (r, x, t) = Tn (r) Pn (x) e−γt , Ψ (r, x, t) = φn (r) Pn (x) e−γt ,
(3.22)
where Pn (x) is the nth Legendre polynomial.
3.2.6
Velocity field governing equations
The rotational field (3.19) satisfies the vorticity equation (3.11c). Substituting the
normal mode (3.22) into (3.11c) generates an equation governing Tn (r)
µ n (n + 1)
µ d2 Tn
+
γT
−
Tn = 0.
n
ρ dr2
ρ
r2
(3.23)
The velocity potential φ is chosen such that the incompressibility condition (3.11a)
is satisfied. Substituting (3.21,3.22) into (3.11a) results in an inhomogeneous equation for ϕn (r)
∇2 (φn (r) Pn (x)) = −
53
1 d ¡ 2 ¢
r Tn Pn (x) .
r2 dr
(3.24)
A general solution for the velocity field (3.20) is constructed by solving (3.23,3.24).
Equations (3.11)–(3.16) form an eigenvalue problem on the interface deformation y(x). Before solving the fully viscous problem, it will be instructive to first
consider the inviscid limit.
3.3
Inviscid solution method
An integro-differential eigenvalue equation, governing the interface deflection, is
derived in this section. To compute the spectrum of eigenvalues and corresponding mode shapes, the integro-differential equation is formulated as a functional
equation on linear operators, which is reduced to a set of linear algebraic equations by a Rayleigh-Ritz variational procedure. The necessary input to such a
procedure is a predetermined function space, which is constructed to satisfy the
no-penetration condition on the surface-of-support and to couple the independent
free surface perturbations, η1 and η2 , according to the incompressibility condition
(3.16). Equivalently, the two free surfaces are allowed to ‘communicate’ across the
spherical-belt constraint through the underlying fluid, subject to the conservation
of volume constraint. The eigenfrequencies/modes are computed using standard
numerical routines from a truncated set of linear algebraic equations.
54
3.3.1
Velocity potential solution
In the inviscid limit, the velocity field is described by the velocity potential, which
satisfies Laplace’s equation
·
1
∇ (φn (r)Pn (x)) = 2
r
µ
2
r
2 dφn
¶
¸
− n (n + 1) φn Pn (x) = 0 [D].
dr
(3.25)
The kinematic condition takes the form
h
i
∂φ
= −γy (x) ∂D1f , ∂D2f ,
∂r
∂φ
vr =
= −γy (x) = 0 [∂Ds ] .
∂r
vr =
(3.26a)
(3.26b)
To satisfy (3.25,3.26), consider the three surfaces as one interface and restrict the
interface perturbations to be functions which vanish on the support (3.3c). Equation (3.26b) will be satisfied by construction for such functions, which are called
‘admissible’. Equations (3.25,3.26) are then recognized as a standard Neumann
type boundary-value problem, whose solution is given by (Arfken & Weber, 2001)
∞
X
d n ³ r ´n
Pn (x),
φ (r, x) = −γR
n R
n=1
µ ¶n+1
∞
X
dn
R
e
φ (r, x) = γR
Pn (x),
n+1 r
n=1
i
(3.27a)
(3.27b)
where
dn ≡
(y, Pn )
.
(Pn , Pn )
(3.28)
Here (f, g) is the inner product of square integrable functions on the domain x ∈
(−1, 1),
Z
1
(f, g) =
f (x)g(x) dx.
−1
55
(3.29)
3.3.2
Pressure
The pressure field, valid for small oscillations, is governed by the Bernoulli equation
p = p0 + ργφ [D],
(3.30)
where p0 is the static pressure required to maintain the undisturbed spherical shape
of the drop. For vanishing viscosity µ, the normal stress boundary condition (3.15)
relates the difference in pressure across the interface to the local curvature there
·
¸
¢
1 ¡
i
e
2
p −p =σ
1 − x yxx − 2xyx + 2y .
(3.31)
R2
3.3.3
Operator equation
Evaluating the pressure from (3.30) at the drop surface and substituting into (3.31)
results in an integro-differential operator equation governing allowable interface
deformations,
"∞ µ
#
¶
2 3 X
¡
¢
1
ρ
γ
R
ρ
1
i
e
1 − x2 yxx − 2xyx + 2y =
+
dk Pk (x) .
σ
k ρi k + 1
k=1
(3.32)
Additionally, the interface deformation y must satisfy the following boundary/integral conditions;
y(±1) − bounded,
Z 1
y(x) dx = 0,
(3.33a)
(3.33b)
−1
y (ζ1 ≤ x ≤ ζ2 ) = 0,
(3.33c)
y(ζ1 ) = 0,
(3.33d)
y(ζ2 ) = 0,
(3.33e)
with ζ1 ≡ cos(θ1 ) and ζ2 ≡ cos(θ2 ).
56
Equations (3.33c–3.33e) enforce the no-penetration condition on the surface-ofsupport, while (3.33b) requires the perturbation to be volume conserving, necessary
for incompressible fluids. Equation (3.33a) guarantees the interface deformation is
physical.
There is no dissipation in inviscid fluids for interfaces that are pinned, therefore
the growth rate γ is purely imaginary. Let γ = iω and define λ2 ≡ ρi ω 2 R3 /σ.
To solve the eigenvalue problem (3.32,3.33), the integro-differential equation is
formulated as an operator equation
K [y] = λ2 M [y] ,
(3.34)
where
¡
¢ d2 y
dy
K [y] ≡ 1 − x
− 2x + 2y
2
dx
dx
2
is a self-adjoint differential operator and
#
"∞ µ
¶
·
¸
X 1 ρe 1 ¶ µ 2k + 1 ¶ µZ 1
ρe
y Pk dx Pk (x)
M y;
≡−
+
ρi
k
ρ
2
i k+1
−1
k=1
(3.35)
(3.36)
is a positive-definite integral operator. Here the density ratio ρe /ρi is a material
parameter.
3.3.4
Rayleigh-Ritz method
The eigenvalue equation (3.34) is posed as a variational one, using Rayleigh-Ritz
formalism, whereby the eigenvalues are computed by minimizing the following
functional,
λ2 = min
(K [y] , y)
, y∈S
(M [y] , y)
(3.37)
over a given function space S. Some details of the theory will be presented here,
while a thorough discussion can be found in e.g. Segel (1987).
57
Given n orthonormal basis functions ψj (x), which span the function space
S, the variational problem is reduced to a set of linear algebraic equations from
which the eigenvalues/vectors are computed. A solution is constructed as a linear
combination of the orthonormal basis functions,
y(x) =
n
X
aj ψj (x).
(3.38)
j=1
Equation (3.38) is applied to the functional (3.37) and minimized with respect to
the coefficients aj , subject to the constraint (y, y) = 1. The resulting set of linear
equations are written as
Kij aj = λ2 Mij aj
with
Z
Z
1
Kij ≡
(3.39)
1
K[yi ]yj dx, Mij ≡
−1
M [yi ]yj dx.
(3.40)
−1
Equation (3.39) is solved using standard numerical techniques. Given an eigenvalue
(k)
λ2(k) and eigenvector aj , the corresponding eigenfunction is
y
(k)
(x) =
n
X
(k)
aj ψj (x).
(3.41)
j=1
3.3.5
Constrained function space
To use the Rayleigh-Ritz procedure on equation (3.34), a function space that satisfies (3.33) is constructed. To begin, consider a piecewise test function




f1 (x) −1 ≤ x ≤ ζ1




f (x) = 0
ζ1 ≤ x ≤ ζ2






f2 (x) ζ2 ≤ x ≤ 1,
58
(3.42)
subject to the following conditions;
Z ζ1
Z
f1 (x) dx +
−1
1
f2 (x) dx = 0,
(3.43a)
ζ2
f1 (ζ1 ) = 0,
(3.43b)
f2 (ζ2 ) = 0.
(3.43c)
By construction, the test function (3.42) satisfies (3.33c) and therefore the nopenetration condition (3.26b). The functions f1 (x) and f2 (x) are the deformations
of the respective free surfaces and are completely independent, except for coupling
via the conservation of volume constraint (3.43a). The perturbation is singlevalued, which dictates that its amplitude must vanish on the boundaries of the
spherical-belt constraint (3.43b,3.43c).
To construct functions that satisfy (3.43), assume the free surface perturbations
take the form;
f1 (x) =
N
X
ck Pk (x), f2 (x) =
N
X
dk Pk (x).
k=0
k=0
Substitution of (3.44) into (3.43) gives
Z ζ1
Z ζ1
Z
c0
P0 dx + · · · + cN
PN dx + · · · + dN
−1
(3.44)
−1
1
PN dx = 0,
(3.45a)
ζ2
c0 P0 (ζ1 ) + c1 P1 (ζ1 ) + · · · + cN PN (ζ1 ) = 0,
(3.45b)
d0 P0 (ζ2 ) + d1 P1 (ζ2 ) + · · · + dN PN (ζ2 ) = 0,
(3.45c)
which is a set of 3 algebraic equations on the coefficients ck and dk

R ζ1
R ζ1
R1
R1
P
(x)dx
·
·
·
P
(x)dx
P
(x)dx
·
·
·
P (x)dx
0
N
0
−1
ζ2
ζ2 N
 −1


P0 (ζ1 )
···
PN (ζ1 )
0
···
0


0
···
0
P0 (ζ2 )
···
PN (ζ2 )



 [c] = [0] .


(3.46)
There are 2(N + 1) − 3 = 2N − 1 linearly independent coefficient vectors that
solve (3.46) and equivalently 2N − 1 linearly independent basis functions ξk (x)
59
which solve (3.43). To use the Rayleigh-Ritz procedure, an orthonormal set of
functions is needed. This step is done using Gram-Schmidt orthogonalization and
a computer algebra package. The orthonormal basis functions inherit the properties of the linearly independent basis functions, such as identically satisfying the
boundary/integral conditions (3.33). Finally, a solution series, which spans the
constrained function space, is constructed using the orthonormal basis functions
ψk (x) as
y(x) =
2N
−1
X
ak ψk (x).
(3.47)
k=1
3.4
Inviscid results
The solution series (3.47) is used to reduce the operator equation (3.34), via a
Rayleigh-Ritz procedure, to a standard eigenvalue problem. The eigenfrequencies/modes, as they depend upon ρe /ρi , ζ1 and ζ2 , are then computed from (3.39).
Setting N = 7 for computation shows eigenvalue convergence to within 0.1% for
the results presented here. Equivalently, 13 terms are used in the solution series
(3.47) with a resolution of 8 terms on each free surface.
The formalism developed here can be specialized to a pinned latitudinal circleof-contact constraint by setting ζ1 = ζ2 . In contrast to the pinned spherical
drop considered in Chapter 2, a broader class of allowable solutions, which are
less restrictive and accommodate discontinuous contact angles across the pinned
circle-of-contact, are considered. Stated differently, in Chapter 2 the problem was
idealized as a single free surface with solutions that necessarily have a continuous
contact-angle across the circle-of-contact, whereas in the present study we treat
the two free surfaces as independent, giving rise to solutions with a discontinu-
60
(a)
Λ
(b)
2
Λ
4
3
2
C2
DC 2
C3
DC 3
60
C1
DC 1
SC
50
40
2
30
20
1
10
0.2
0.4
0.6
0.8
1
Ζ1
0.2
0.4
0.6
0.8
1
Ζ1
Figure 3.2: Frequency (λ2 ) comparison between continuous (C), discontinuous (DC) and spherical cap (SC) perturbations for a pinned
circle-of-contact (ζ1 = ζ2 ) with (a) n = 1 (b) n = 2, 3 .
ous contact-angle. As seen in figure 3.2, which plots the eigenfrequency λ2 vs.
pin location ζ1 for ρe /ρi = 0, the continuous contact-angle frequencies (C) are always larger than the corresponding discontinuous frequencies (DC), with equality
achieved at local minima. Here the frequencies are symmetric with respect to the
pin location, λ2n (ζ) = λ2n (−ζ), and the value of the local minima correspond to
the ‘unconstrained’ Rayleigh frequencies (2.1) for mode number n. These minima are termed ‘natural’ pin locations and specified by the ‘nodes’ or zeroes of
the corresponding unconstrained mode shapes. As seen from the computed mode
shapes of figures 3.3(a, c), the equator is a natural pin location for the odd mode
shapes. Likewise, a pin location out-of-phase with respect to the unconstrained
mode shapes is analogously called ‘unnatural’ and may be associated with the local maxima of figure 3.2. For example, consider the n = 2 mode shape shown in
figure 3.3(b). This mode clearly shows a discontinuous contact-angle for a drop
pinned at the equator. Using the variational problem interpretation, one could say
a perturbed liquid drop would generally prefer to oscillate with a discontinuous
contact-angle across the pinned circle-of-contact.
61
(a)
(b)
(c)
Figure 3.3: Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 for a drop with
pinned circle-of-contact located at the equator (ζ1 = ζ2 = 0).
One goal in generalizing to disjoint free surfaces with discontinuous contact
angles is to compare the low-frequency n = 1 mode to the spherical cap model
of Theisen et al. (2007). In the preceeding chapter, a snapshot is provided of the
corresponding experiments to clearly show the different apparent contact angles
across the constraint. Figure 3.2(a) compares the spherical cap model (SC) with
the n = 1 frequency from the continuous (C) and discontinuous (DC) contact-angle
analysis. As shown, the qualitative behavior is similar in each analysis, but the
discontinuous model most closely resembles the spherical cap model, which can be
directly attributed to the larger class of solutions for this model.
The spherical-cap behavior of the n = 1 mode shape, shown in figures 3.4(a)
and 3.5(a), persists for finite-sized spherical-belt constraints. Additionally, the
n 6= 1 mode shapes of figures 3.4,3.5 show rich behavior, illustrating different ways
that the two free surfaces communicate across the constraint. For example, the
mode shapes of figures 3.4(b),3.4(c) and 3.5(b) display spherical-cap behavior on
one free surface and higher-order shapes on the other. This leads one to believe that
eigenmodes may be excited where one free surface is relatively ‘inactive’ compared
to the second free surface. Alternatively, figure 3.5(c) demonstrates that higher62
(a)
(b)
(c)
Figure 3.4: Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 with spherical-belt
constraint ζ1 = −0.7, ζ2 = 0.
(a)
(b)
(c)
Figure 3.5: Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 with spherical-belt
constraint ζ1 = −0.6, ζ2 = 0.8.
order shapes can occur on both free surfaces.
Strani & Sabetta (1984) have shown that increasing the size of their sphericalbowl constraint leads to exponential growth of the eigenfrequency. To examine
the effect of constraint size, the first five eigenfrequencies are plotted in figure 3.6
as a function of the second pin location ζ2 , while holding the first pin location
ζ1 fixed. The frequency growth is monotonic with constraint size, but there are
certain well-defined plateaus or ‘dead zones’, where a further increase in belt-size
has no substantial influence on the frequencies themselves. In these regions, no ap63
Λ2
n‡1
n‡2
n‡3
n‡4
n‡5
400
300
200
100
C
B
A
0.5
0.6
0.7
0.8
0.9
Ζ
1 2
Figure 3.6: Frequency λ2 against second pin location ζ2 , while holding ζ1 =
0.4.
(a)
(b)
(c)
Figure 3.7: Mode shape (n = 3) at (a) point A(ζ1 = 0.4, ζ2 = 0.45) (b)
point B(ζ1 = 0.4, ζ2 = 0.85) (c) point C(ζ1 = 0.4, ζ2 = 0.95) of
Figure 3.6.
preciable change in mode shape is observed. Alternatively, one might say that one
belt-size appears to be no more desirable than the next. As such, figures 3.7(a, b)
show only a slight variation in the n = 3 mode shape, despite the drastic difference in constraint size.
Figure 3.6 also displays accelerated eigenfrequency
growth in particular regions. In these regions, one surface seems relatively inactive, while the other is actively changing its character (e.g. number of nodes) to
64
(a)
(b)
1.0
1.0
2
1
0.5
Ζ2
1
0.5
0
Ζ2
0.0
2
0.0
1
1
-0.5
-0.5
2
-1.0
-1.0
-0.5
0.0
0.5
-1.0
-1.0
1.0
0.0
-0.5
Ζ1
(c)
(d)
0.0
1.0
3
2
Ζ2
1.0
Ζ1
1.0
3
3
0.5
0.5
3
0.5
2
Ζ2
2
4
3
3
0.0
3
4
-0.5
4
-0.5
3
3
3
-1.0
-1.0
-0.5
0.0
0.5
1.0
Ζ1
-1.0
-1.0
-0.5
0.0
0.5
Ζ1
Figure 3.8: Geometric index (engineering) of mode (a) n = 1, (b) n = 2, (c)
n = 3 and (d) n = 4, against ζ1 and ζ2 .
65
1.0
(a)
(b)
Figure 3.9: Geometric index: comparison between the engineering and mathematical interpretation.
fit the given constraint. These changes are tracked in figure 3.7, which plots the
n = 3 mode shape at points A, B, C of the n = 3 curve from figure 3.6. The transition from figure 3.7(b) to figure 3.7(c) occurs when the n = 2 and n = 3 curves
of figure 3.6 cross. More definitively, such a point is identified mathematically
by the eigenvalue’s algebraic and geometric multiplicity of two and one, respectively. The defining feature of the coalescence point is qualitative and associated
with a change in the number of nodes of the corresponding mode shape, despite
preserving the numerical ordering of eigenvalues. One can monitor node-creation
or-destruction by defining a geometric index of a given mode shape as the number
of times the disturbed shape intersects the undisturbed shape. The boundaries
of the spherical-belt constraint are not included in this definition, which will be
referred to as the ‘engineering index’. The index can be defined on the half-drop,
because of the axisymmetry assumption (c.f. figure 3.9). The engineering index,
as it depends upon the geometry of the spherical belt, is shown in figure 3.8 for
66
the first four modes. In unconstrained problems, the spectral ordering of eigenvalues and number of nodes of the respective mode shape are coincident, as for
the unconstrained Rayleigh modes. The constrained nature of the problem considered here distorts this ordering, whereby the mode shape associated with the third
numerical eigenvalue may have less than three nodes for a fixed constraint. The
ambiguity in ordering occurs at the coalescence points of figure 3.6 and is tracked
using the engineering index in figure 3.8. To reconcile this ambiguity, one can
define a new geometric index to account for the possibility of a virtual node within
the spherical-belt constraint. This interpretation is termed the ‘mathematical index’. Specifically, if the amplitude of the linear disturbance has opposite sign on
either side of the spherical-belt, continuity requires the existence of an additional
‘virtual’ node within the spherical-belt constraint (c.f. figure 3.9(b)). In this case,
the mathematical index is simply the engineering index plus one. Alternatively,
when the disturbance has the same sign on both sides of spherical-belt then there
does not exist a virtual node and the engineering and mathematical index are coincident (c.f. figure 3.9(a)). The mathematical index is identical to the numerical
order of the respective frequency and is thus unambiguous with respect to the
classical ordering of mode shapes. In addition, the mathematical index reduces to
the index of the Rayleigh modes in the limit of a pinned circle-of-contact (ζ1 = ζ2 ).
Another distinguishing feature of figure 3.6 is that horizontal asymptotes connect dead regions of different mode number. The asymptotes are lines of constant
λ2 , whose numerical value is given by the corresponding eigenfrequency for the
drop constrained by a spherical bowl (Strani & Sabetta, 1984), whose size 1 − ζ1
is set by fixing ζ1 and taking the limit ζ2 → 1. Furthermore, if one connects each
successive point of coalescence, the resulting curve is well-defined and exponential.
As this curve is traversed, one can monitor node creation.
67
(a)
(b)
Vn
Vn
0.6
n‡1
n‡2
n‡3
0.5
0.4
D12
0.5
0.4
0.3
0.3
D23
0.2
0.2
D23
0.1
0.3
0.4
0.5
0.6
0.7
0.1
0.8
0.9
1
Ζ2
0.5
0.6
(c)
Λ
n‡1
n‡2
n‡3
D12
Λ
0.9
Ζ
1 2
0.9
Ζ
1 2
2
150
n‡1
n‡2
n‡3
150
0.8
(d)
2
200
0.7
n‡1
n‡2
n‡3
D23
100
D23
100
50
50
D12
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ζ2
D12
0.5
0.6
0.7
0.8
Figure 3.10: Perturbed volume Vn (a, b) and frequency λ2 (c, d) vs. ζ2 for
fixed (a),(c) ζ1 = 0.2 and (b),(d) ζ1 = 0.4.
Physically, these crossings may be located by examining the perturbed volume
exchange between disjoint interfaces for a given mode shape y (n) , defined as
Z 1
Vn ≡
y (n) (x) dx.
(3.48)
ζ2
Figures 3.10(a), (b) plot the perturbed volume exchanged Vn as a function of the
second pin location ζ2 , while holding the first pin location ζ1 fixed. From these
figures, one can identify D12 and D23 as crossing points, where the primary volumecarrying mode-shape changes from n = 1 to n = 2 and n = 2 to n = 3, respectively.
As shown in figures 3.10(c, d), which plots frequency λ2 vs. ζ2 for fixed ζ1 , such
points are precisely the coalescence points, which were previously defined mathematically. To summarize, at transition point Dn(n+1) the n + 1 mode shape adds
68
another node and becomes the primary volume carrier, taking volume from the n
mode shape.
3.4.1
Perturbed-volume embedding
To further investigate the role of volume-exchange amongst disjoint interfaces, consider an arbitrary domain, bound by two free surfaces and subject to the following
constraint
Z
Z
∂D1f
y1 = C,
∂D2f
y2 = −C.
(3.49)
Here, a perturbed volume C is exchanged between interfaces, while total volume
is conserved, in accordance with the assumption of fluid incompressibility. As
the governing equations can be derived from a variational approach, one may
incorporate the constraints (3.49) into an augmented functional
Z
F [y1 , y2 ; µ1 , µ2 ] =
(λ1 M [y1 ] + K [y1 ] + µ1 ) y1
∂D1f
Z
+
(λ2 M [y2 ] + K [y2 ] + µ2 ) y2
(3.50)
∂D2f
through the Lagrange multipliers µ1 , µ2 . This formulation is completely equivalent to the solution method employed previously. It utilizes an embedding to
enforce incompressibility, whereas previously this condition was explicitly satisfied
through the construction of the constrained function space. As mentioned above,
the surface disturbance may be defined in a piecewise manner by decomposing
the bounding surface into the union of the free surfaces and surface-of-support,
∂D ≡ ∂D1f ∪ ∂D2f ∪ ∂Ds . Using this decomposition, in conjunction with a continuity condition on the surface-of-support, one can independently vary the free
surface disturbances y1 , y2 , giving rise to the form of (3.50). Stationary values
of the augmented functional are solutions of the corresponding Euler-Lagrange
69
equations
λ1 M [y1 ] + K [y1 ] + µ1 = 0, λ2 M [y2 ] + K [y2 ] + µ2 = 0,
(3.51)
where the Lagrange multipliers may be interpreted physically as the constant pressure applied at each respective interface, requisite to displace a volume C. To
enforce the constraint (3.49) (i.e. embed the perturbed volume), integrate the
Euler-Lagrange equations (3.51) over the respective undisturbed free surface to
specify µ1 , µ2 . The following Euler-Lagrange equations result,
k1 [y1 ] = −λ1 m1 [y1 ; C] , k2 [y2 ] = −λ2 m2 [y2 ; C] .
(3.52)
The structure of the governing equations is reminiscent of a system of weaklycoupled oscillators and shows that the respective interfaces oscillate with an effective spring constant
R
k1 [y1 ] ≡ K [y1 ] −
∂D1f
R
M −1 [K [y1 ]]
∂D1f
M −1 [1]
R
, k2 [y2 ] ≡ K [y2 ] −
∂D2f
R
M −1 [K [y2 ]]
∂D2f
M −1 [1]
, (3.53)
and effective mass, parameterized by C and given by
m1 [y1 ; C] ≡ M [y1 ] − R
∂D1f
C
C
, m2 [y2 ; C] ≡ M [y2 ] + R
. (3.54)
−1
M [1]
M −1 [1]
∂Df
2
One may independently compute the oscillation frequencies λ1 (C), λ2 (C), as they
depend upon the perturbed volume C, from (3.52). However, the interfaces are
coupled through the fluid domain, which requires the frequencies to be in phase,
λ = λ1 (C ∗ ) = λ2 (C ∗ ), at a critical perturbed volume C ∗ . Accordingly, each
mode shape exchanges a characteristic perturbed volume C ∗ between disjoint free
surfaces.
The perturbed-volume embedding can be used to provide useful information
regarding the eigenvalue degeneracy mentioned earlier. Namely, by specifying the
perturbed-volume makes the eigenvalue problem well-posed, as opposed to the
70
under-specified standard two-point boundary-value problem. Accordingly, mode
shapes with the same perturbed volume are necessarily identical by uniqueness,
giving rise to degenerate eigenvalues, as shown in figures 3.10(a), (b). Additionally,
the perturbed volume C can be viewed as a measure of ‘communication’ between
free surfaces. A non-trivial perturbed volume implies the disjoint surfaces are
in communication, while mode shapes with zero perturbed volume exchange are
essentially de-coupled.
3.5
Viscous solution method
The equation governing the interface deflection of a viscous drop is derived and
formulated as an operator equation in this section. The no-penetration condition
(3.12b) is satisfied by restricting the interface perturbations to the constrained
function space derived previously.
However, the ‘shear’ boundary conditions
(3.12a),(3.13a) and (3.13b) change from no-stress (free) to no-slip (supported).
To address this issue, a modified set of boundary conditions are proposed. A normal mode analysis with the modified boundary conditions allows one to derive the
viscous operator equation, which depends upon a viscosity parameter ², the ratio
of inner to exterior densities ρ, the ratio of inner to exterior viscosities µ, and a
support size parameter. The operator equation is solved for both a viscous drop
in a vacuum and an inviscid fluid (bubble) immersed in a viscous liquid. A general
operator equation for an immiscible viscous drop is given in Appendix B. Unlike
the inviscid limit, the eigenvalues are computed from a characteristic equation with
eigenvalues appearing nonlinearly.
71
3.5.1
Vector potential solution
The vector potential solution of (3.23) is given by
Tni (r)
=
³ r ´1/2
R
³ r ´1/2
Hn+1/2 (z e )
Jn+1/2 (z i )
e
e
, Tn (r) =
Tn (R) (1)
,
Jn+1/2 (Z i )
R
H
(Z e )
(1)
Tni (R)
(3.55)
n+1/2
where
µ
z
i,e
ρi,e
≡r γ
µi,e
¶1/2
, Z
i,e
µ
¶1/2
ρi,e
≡R γ
,
µi,e
(3.56)
(1)
and Jn (z) and Hn (z) are the appropriate Bessel functions (standard notation
used, Arfken & Weber, 2001).
3.5.2
Velocity potential solution
Equation (3.24) is solved using variation-of-parameters for the velocity potential,
·
¸
Z r i,e
n+1
Tn (s)
i,e
i,e
φn (r) = αn −
ds rn
2n + 1 R sn
¸
·
(3.57)
Z r
n
n+1 i,e
−(n+1)
i,e
s Tn (s) ds r
.
+ βn −
2n + 1 R
The velocity potential is finite as r → 0 and r → ∞, which requires
βni
n
=−
2n + 1
Z
R
s
0
and
αne
n+1
n+1
=
2n + 1
Z
∞
R
Tni (s) ds
n
Tni (R)
=−
Rn+2
i
2n + 1 Jn+3/2 (Z )
n+1
Tne (s)
Tne (R)
ds
=
R−(n−1) ,
n
e
s
2n + 1 Hn−1/2 (Z )
(3.58)
(3.59)
where
(1)
H
(z)
Jn−1 (z)
Jn (z) ≡ z
, Hn (z) ≡ z n+1
(1)
Jn (z)
Hn (z)
(3.60)
are fractional Bessel functions (again standard notation, Arfken & Weber, 2001).
72
3.5.3
Pressure
The pressure is found by substituting (3.22) into (3.11b) to give
µ
¶
µ dTn
Pn (x).
p = p0 + ρ γφn −
ρ dr
(3.61)
Using the vector and velocity potential solutions (3.55,3.57), the pressure evaluated
on the surface is
·
¸
Tni (R) ρi Rγ 2
p (R) = − (n + 1) µi
+
dn Pn (x),
R
n
·
¸
Tne (R) ρe Rγ 2
e
e
p (R) = p0 + (n)µe
+
dn Pn (x).
R
n+1
i
3.5.4
pi0
(3.62a)
(3.62b)
Boundary conditions
As with the other field quantities, the interface perturbation is expanded as
y(x) = dn Pn (x), dn ≡
(y, Pn )
.
(Pn , Pn )
(3.63)
Using (3.21) and (3.63), the linearized kinematic condition (3.14) is reduced to
Tni (R) +
dφin ¯¯
dφe ¯¯
= −γdn , Tne (R) + n ¯
= −γdn .
¯
dr r=R
dr r=R
(3.64)
As in the inviscid limit, equation (3.64) is valid on the entire interface, and the nopenetration condition (3.12b) is satisfied by restricting candidate functions to an
appropriately chosen function space. To simplify (3.64), use the velocity potential
solution (3.57) to obtain
αni =
n + 1 −(2n+1) i γ −(n−1)
n
γ
R
βn − R
dn , βne =
R2n+1 αne +
Rn+2 dn . (3.65)
n
n
n+1
n+1
with βni and αne given by (3.58) and (3.59), respectively.
73
1
f
¶D
f
¶D1
-1
¶D2
s
Ζ1
Ζ2
1
x
Figure 3.11: Indicator function Γ(x, ζ1 , ζ2 )
The remaining unknowns, Tni (R) and Tne (R), are found from the shear boundary conditions; (3.12a),(3.13a) and (3.13b). These boundary conditions are valid
on specified parts of the interface and are not amenable to standard analysis.
To resolve this issue, a new set of boundary conditions are proposed. On the
drop interface, the following boundary conditions are introduced as equivalent to
(3.12a),(3.13a) and (3.13b);
vθi |r=R = vθe |r=R ,
¡ i
¢
e
τrθ − τrθ
|r=R [1 − Γ (x, ζ1 , ζ2 )] = CΓ (x, ζ1 , ζ2 ) vθi |r=R .
(3.66a)
(3.66b)
Here Γ(x, ζ1 , ζ2 ) is an indicator function, which is active on the surface-of-support
and inactive on the free surfaces (c.f. figure 3.11),
Γ(x, ζ1 , ζ2 ) ≡ H(x − ζ1 ) − H(x − ζ2 ),
(3.67)
where H(x) is the Heaviside step function. The constant C, which has units of
[viscosity]/[length], will be determined when the problem is scaled.
74
3.5.5
Viscous operator equation
The viscous operator equation is derived from the normal stress boundary condition
(3.15)
·
¸
¢
¢
∂vre ¯¯
1 ¡¡
∂vri ¯¯
e
2
+ p − 2µe
= −σ
1 − x yxx − 2xyx + 2y ,
−p + 2µi
¯
¯
∂r r=R
∂r r=R
R2
i
which can be written as
·µ
¶
¸
γ
ρe
ρi
2
γ R + 2 (n − 1) (n + 2) (µi − µe )
dn
+
n+1
n
R
T e (R)
T i (R)
+n (n + 2) µe n
− (n − 1) (n + 1) µi n
R
R
£¡
¢
¤
= −σ/R2 1 − x2 yxx − 2xyx + 2y ,
(3.68)
with Tni (R) and Tne (R) determined from the modified boundary conditions (3.66).
This is done in Appendix B.
The following scalings are introduced,
r
ρi R 3
µi,e
µe
ρe
∗
γ, ²i,e ≡ p
, µ≡ , ρ≡ .
γ ≡
σ
µi
ρi
ρi,e Rσ
A product of the modified boundary conditions is a ‘shape’ factor
¶µ
µZ 1
¶
¡ (1) ¢2
2n + 1 (n − 1)!
Pn (x) Γ(x, ζ1 , ζ2 ) dx
Ln =
,
2 (n + 1)!
−1
(3.69)
(3.70)
(1)
which is a measure of the relative size of the surface-of-support. Here Pn (x) is
the nth Legendre polynomial of order one (MacRobert, 1967).
Drop in vacuum
A viscous drop in a vacuum corresponds to the limiting case µe → 0 and ρe → 0.
In this limit, the viscous drop operator equation is given by
γ ∗ 2 Md [y] + γ ∗ Φd [y; γ ∗ , ²i ] + K [y] = 0.
75
(3.71)
The differential operator
¡
¢
K [y] ≡ 1 − x2 yxx − 2xyx + 2y
(3.72)
is associated with the curvature and acts as a restoring force (same as equation
(3.35)), while the positive-definite inertia operator is defined as
Md [y] ≡
X1
dn Pn (x).
n
n=1
(3.73)
Viscous effects are controlled by the dissipation operator
Φd [y; γ ∗ , ²i ] ≡ −²i
X£
¤
2 (n − 1) (n + 2) + (n − 1) (n + 1) Tni (R) dn Pn (x) (3.74)
n=1
where
"
2 (n − 1) /n − ²i An /n
¢
Tni (R) = ¡
2/Jn+3/2 (X i ) − 1 + ²i An /Jn+3/2 (X i )
#
(3.75)
and
An ≡ Ln /(1 − Ln ); X i ≡ (γ ∗ /²i )1/2 .
(3.76)
Bubble in viscous medium
A similar operator equation is derived for a bubble, the limit µi → 0 and ρi → 0
in equation (3.68),
γ ∗ 2 Mb [y] + γ ∗ Φb [y; γ ∗ , ²e ] + K [y] = 0
(3.77)
where the curvature operator is defined in (3.72) and the inertia operator for a
bubble as
Mb [y] ≡
X
n=1
1
dn Pn (x).
n+1
(3.78)
The bubble dissipation operator is given by
Φb [y; γ ∗ , ²e ] ≡ −²e
X
[−2 (n − 1) (n + 2) + n (n + 2) Tne (R)] dn Pn (x).
n=1
76
(3.79)
with
"
2(n + 2)/(n + 1) − ²e An /(n + 1)
¢
Tne (R) = − ¡
2/Hn−1/2 (X e ) − 1 + ²e An /Hn−1/2 (X e )
#
(3.80)
and
An ≡ Ln /(1 − Ln ); X e ≡ (γ ∗ /²e )1/2 .
(3.81)
Immiscible viscous drop in viscous medium
The operator equation for an immiscible viscous drop is given in Appendix B.
3.5.6
Viscous operator solution
The operator equations (3.71) and (3.77) are nonlinear in the eigenvalue and reduced to a set of algebraic equations using a Rayleigh-Ritz procedure with the function space (3.47) derived in section 3.3.5. The resulting matrix equation is parameterized by the viscosity parameter ² and the boundaries of the belted constraint,
ζ1 and ζ2 , via the indicator function Γ(x, ζ1 , ζ2 ). The eigenvalues/eigenmodes are
then computed from a nonlinear characteristic equation, found by taking the determinant of the matrix equation.
3.6
Viscous results
The viscous eigenvalues/modes for the drop and bubble, as they depend upon
the viscosity parameter ², are the roots of a nonlinear characteristic equation and
computed using a variant of the secant method. As with the inviscid limit, 13
terms are used in the solution series (3.47), with a resolution of N = 7 Legendre
77
polynomials on each free surface, to derive the characteristic equations from the
Rayleigh-Ritz procedure. As the viscosity parameter ² is varied, the eigenvalues
bifurcate from complex to strictly real. The motion transitions from under-damped
to over-damped.
3.6.1
Checks on viscous solution
A number of checks are performed to verify both the numerical routine as well
as the modified boundary conditions (3.66). First, consider the limiting case of a
pinned circle-of-contact constraint, ζ2 → ζ1 . In this limit, the ‘shape’ factor (3.70)
tends to zero and the operator equations (3.71,3.77) can be manipulated into the
functional equivalent of the Prosperetti (1980b) equations for the unconstrained
drop. The modified boundary conditions (3.66) are equivalent to the boundary conditions for an unconstrained drop, provided the no-penetration condition (3.12b)
is satisfied, which can be accomplished by constructing the appropriate function
space. The no-slip condition cannot be satisfied in this limit, which implies ‘slip’
across the circle-of-contact.
A second check is performed in the limit ζ1 → −1, where the spherical-belt
constraint is equivalent to the spherical-bowl constraint analyzed by Strani & Sabetta (1988). Figure 3.12 plots the real and imaginary parts of the first three
eigenfrequencies as a function of the viscosity parameter ²i for a drop in a vacuum
with spherical-bowl constraint. The n = 2, 3 eigenvalues bifurcate from complex
to real at a critical value of the viscosity parameter, which is smaller for the n = 3
mode as compared with the n = 2 mode, because there is stronger relative motion
between fluid elements as the mode number increases. As can be seen from figure 3.13, which plots the viscous frequencies for a bubble pinned on the south pole
78
(a)
(b)
*
*
Im@Γ D
Re@Γ D
n‡1
n‡2
n‡3
14
12
10
n‡1
n‡2
n‡3
8
6
8
4
6
4
2
2
0.2
0.4
0.6
0.8
Ε
1.0 i
0.2
0.4
0.6
0.8
Ε
1.0 i
Figure 3.12: Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ]
against viscosity parameter ²i for a viscous drop with sphericalbowl support (ζ1 = −1, ζ2 = −0.8).
(a)
(b)
*
*
Re@Γ D
Im@Γ D
4
8
n‡1
n‡2
n‡3
3
2
4
1
2
0.0
0.5
1.0
1.5
n‡1
n‡2
n‡3
6
Ε
2.0 e
0.5
1.0
1.5
Ε
2.0 e
Figure 3.13: Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ]
against viscosity parameter ²e for a bubble pinned at the south
pole (ζ1 = −1, ζ2 = −0.99).
(ζ1 = −1, ζ2 = −0.99), the transition from under-damped to over-damped motion
does not occur for the bubble. Figures 3.12 and 3.13 are in excellent agreement
with the results of Strani & Sabetta (1988).
Finally, a spherical-belt constraint is chosen and the complex frequencies are
computed for the drop and bubble in figures 3.14 and 3.15, respectively. For
reference, the drop mode-shapes for this constraint are plotted in figure 3.16. The
79
(a)
(b)
*
*
Im@Γ D
Re@Γ D
12
n‡1
n‡2
n‡3
10
8
n‡1
n‡2
n‡3
8
6
6
4
4
2
2
0.2
0.4
0.6
0.8
Ε
1.0 i
0.2
0.4
0.6
0.8
Ε
1.0 i
Figure 3.14: Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ]
against viscosity parameter ²i for a drop with spherical-belt support (ζ1 = −0.2, ζ2 = 0.4).
(a)
(b)
*
*
Re@Γ D
Im@Γ D
5
10
n‡1
n‡2
n‡3
4
3
6
2
4
1
2
0.5
1.0
1.5
n‡1
n‡2
n‡3
8
Ε
2.0 e
0.5
1.0
1.5
Ε
2.0 e
Figure 3.15: Decay rate (a) Re[γ ∗ ] and oscillation frequency (b) Im[γ ∗ ]
against viscosity parameter ²e for a bubble with spherical-belt
support (ζ1 = −0.2, ζ2 = 0.4).
dissipation arising from relative fluid motion is very apparent when one compares
the growth rate of the n = 1 mode to the n = 2, 3 modes in figure 3.14, where
the magnitude is much more pronounced for large mode numbers. To illustrate,
consider figure 3.16 which shows very little relative motion for the n = 1 mode
shape as compared with the n = 2, 3 modes.
80
(a)
(b)
(c)
Figure 3.16: Mode shapes (a) n = 1 (b) n = 2 (c) n = 3 for a drop with
spherical-belt support (ζ1 = −0.2, ζ2 = 0.4).
3.7
Concluding remarks
The linear oscillations of an immiscible, viscous fluid drop, held by surface tension and supported with a spherical-belt constraint has been considered here. The
integro-differential equation governing the interface deformation is formulated as
an eigenvalue problem on linear operators. A solution is generated using the variational procedure of Rayleigh-Ritz on a constrained function space. To construct
the constrained function space, one can define the interface as the union of the two
free and one surface-of-support and restrict ‘allowable’ solutions to appropriately
chosen candidate functions. At this level of generality, the free surface deformations are independent and allowed to communicate, or exchange volume through
the region beneath the surface-of-support, but coupled by a conservation of volume
constraint.
The formalism developed here treats the two free surfaces independently and
allows one to seek solutions in a broad class of functions, giving rise to the possibility of a discontinuous contact-angle across a pinned circle-of-contact constraint, in
contrast to Chapter 2, where the contact-angle is necessarily continuous. Compar81
ing the inviscid eigenfrequencies from both studies reveals the drop typically has
a discontinuous contact-angle (c.f. figure 3.2). A comparison is made between the
spherical-cap model of Theisen et al. (2007) and the n = 1 ‘slow’ frequency mode
shown figure 3.2(a), where the quantitative agreement is improved by generalizing
to discontinuous contact angles.
As the size of the spherical-belt constraint is increased, there are mode crossings, as shown in figure 3.6. These are multiplicities, where two mode share the
same frequency. Physically, it has been shown that a transition happens, where
the primary volume-carrying mode-shape changes. At these critical belt sizes, the
numerical ordering of the eigenvalues is preserved, but the number of ‘nodes’ or
engineering index of the corresponding mode shape changes, in contrast to the
classical theory of unconstrained linear operators, where these are coincident. The
engineering index, as it depends upon the geometry of the spherical-belt constraint, is shown in figure 3.8. Mathematically, crossings are characterized by an
eigenvalue with algebraic and geometric multiplicity of one and two, respectively.
Clearly, the presence of the constraint distorts the classical ordering of modes by
their index. To address this, an alternative definition of the index is used, whereby
there is the possibility of a ‘virtual’ node within the spherical-belt constraint. The
mathematical index reduces to the Rayleigh index.
A perturbed volume embedding is used in conjunction with the variational
form of the governing equations to provide useful information with regards to the
eigenvalue degeneracy. Namely, it has been shown that coupled, disjoint interfaces oscillate with an effective mass, similar to a system of weakly coupled linear
oscillators. Additionally, mode-shapes with equal perturbed-volume exchange correspond to the eigenfrequency degeneracy.
82
The functional analytic approach used for the inviscid drop has also been used
in formulating the viscous operator equation. A set of modified boundary conditions (3.66), valid on the entire interface, were introduced and validated. An
indicator function (3.67) is used to track the transition from no-shear to no-slip
along the interface. Introduction of the modified boundary conditions allows one
to derive the operator equations (3.71) and (3.77), which are reduced to a set of
linear algebraic equations and give rise to a nonlinear characteristic equation for
the complex eigenvalue γ ∗ . The eigenvalue solutions bifurcate from complex to real
solutions, or under-damped to over-damped motion, at a critical value of the viscosity parameter ² for a viscous drop. No such qualitative behavior change occurs
for the bubble in a viscous medium and under-damped periodic motion persists.
The higher mode-number solutions have a smaller critical viscosity parameter, because these mode shapes have stronger relative motion among fluid elements and
equivalently stronger viscous dissipation. As the size of the spherical-belt shrinks,
it has been shown that ‘slip’ occurs across the pinned circle-of-contact. The modified boundary conditions and the results presented here have been validated against
previous literature (Strani & Sabetta, 1988; Prosperetti, 1980b).
While this work focused on non-trivial interface dynamics of a viscous drop,
there exists shear/rotational wave solutions of the governing equations, if one would
consider an additional vector potential generating a radial component of vorticity.
The extension to these aperiodic motions is straightforward and the methodology
is sketched in Appendix C.
83
CHAPTER 4
STABILITY OF CONSTRAINED CYLINDRICAL INTERFACES
AND THE TORUS-LIFT OF PLATEAU-RAYLEIGH
The content of this chapter has been largely disseminated in Bostwick & Steen (J.
Fluid Mech. (2010), vol. 647, pp. 201-219)
4.1
Introduction
A cylindrical interface holds an underlying liquid in static equilibrium and is subject to dynamic capillary instability, including Plateau-Rayleigh break-up. Plateau
(1863) found instability of the liquid cylinder for lengths longer than the base-state
circumference by comparing surface areas of both the disturbed and undisturbed
base-state shapes, the well-known Plateau limit. Plateau went on to incorrectly
interpret this limit as predicting the size of drop resulting from the instability.
Extending the work of Plateau, Lord Rayleigh (1879) formulated the linearized
hydrodynamic equations and calculated the growth rate as it depends on disturbance wave-number and correctly interpreted the wave-number of maximum growth
rate from this dispersion relation as giving a good approximation to the final drop
size. The Plateau limit is recovered from the Rayleigh result by putting growth
rate to zero. Probably for this reason, the Plateau limit is sometimes referred to
as the Plateau-Rayleigh limit.
The stability and/or dynamics of constrained cylindrical interfaces are important in a number of applications including coating and casting processes, lowgravity liquid containment, and two-phase heat transfer. For example, in planarflow melt spinning molten metal is held by surface tension between a stationary
84
nozzle and a moving substrate (Steen & Karcher, 1997). The upstream meniscus is nearly cylindrical and is part of a full meniscus which takes the shape of
a distorted torus. In fact, Rayleigh oscillations of an inviscid sphere capture the
wavelength of a defect frozen into the ribbon-product (Byrne et al., 2006). The
constrained toroidal geometry used here is potentially relevant to the drop impact
problem of Renardy et al. (2003), where a toroidal rim, attached to a thin liquid sheet is formed after impact. Similarly, the formation of Edgerton crowns, or
the ‘crown-splash problem, result from instability of toroidal rims (Deegan et al.,
2008).
The stability of cylindrical interfaces constrained by wetting contact with a
cylindrical cup of circular cross-section is considered here, figure 4.1. The interface is pinned at contact-lines along the edges of the solid support. The polar
angle of contact θs defines the extent of the cylindrical-cup constraint. First, using
the dynamic approach for inviscid liquids, the modes of instability and dispersion
relations are calculated for 0 < θs < 2π, extending previous results obtained by
the static approach, much as Rayleigh extended the Plateau result. Next, the
cylindrical interface is bent in-plane and required to wet a toroidal-cup constraint,
thereby introducing a weak secondary curvature. That is, toroidal interfaces near
the cylindrical limit are considered. Unlike the cylinder, the torus is not an equilibrium shape but there exists a near-torus equilibrium shape whose cross-sections
are deformed circular arcs. Using a perturbation approach, the static stability of
these base-states is calculated, showing that the constraint can either stabilize or
destabilize depending on its inside or outside position. Moreover, in the case of stabilization, a base-shape is identified for which the destabilization of the secondary
curvature just cancels the stabilization by the wire constraint. That is, the Plateau
limit is lifted to the torus. Finally, using the static approach on the torus-lift of
85
the cylinder, it will be shown that θs = π bounds a stability window, where here
stability is to finite-amplitude disturbances. The argument uses a symmetrization
procedure that goes back to Steiner (1882).
It is well-known that constraints can dramatically influence the stability of
the Plateau-Rayleigh problem, which is relevant to capillary jets, liquid bridges,
columns, ridges and fillet beads, among others. There exists a large volume of literature focused on the static stability of capillary interfaces and the most relevant
to this study are mentioned here. Majumbar & Michael (1976) obtain the static
stability of a cylindrical meniscus pinned along a horizontal slot, while studying
pendant drops. Brown & Scriven (1980) consider the exactly-cylindrical fillet for
constant-pressure as well as constant-volume disturbances and report static stability. Langbein (1990) treats the static stability of cylindrical interfaces constrained
to wet a V-groove, considering liquid in positions both inside and outside the
groove. Roy & Schwartz (1999) also study liquids that partially wet cylindrical
‘containers’ of a variety of cross-sectional shapes, including planar, V-groove, circular, and elliptical. Their static stability analysis of the circular-arc cross-section
base-state recover results of the previous studies and also treat some new substrate geometries. May & Lowry (2008) propose helical and double-helical wire
constraints (pinning locus) to support liquid columns and calculate the static stability of resulting interfaces, which may or may not be nearly-cylindrical. Other
static stability results can be found in Michael (1981), who has reviewed the static
approach.
Despite the fact that static stability results can be recovered from the dynamic
approach, the existing literature with regards to dynamic stability calculations
is sparse. One such example is given by Myshkis et al. (1987) (Section 5.3.4),
86
who sketches one solution approach for vibrations of an ideal liquid underlying
a cylindrical meniscus supported by a V-groove with convex circular bottom, all
for natural boundary conditions (contact-angle of π/2). No dispersion relations
or eigenmodes are presented, but static results are recovered: axial disturbances
are unstable at half the Plateau-Rayleigh limit and planar disturbances for basestate interfaces greater than semi-circular. In a study of moving contact-lines and
rivulets, Davis (1980); Weiland & Davis (1981) considers the stability of a static
rivulet of circular-arc cross-section on a horizontal plate. The Davis (1980) study of
static rivulets conveniently frames this study even though the moving contact-line
is not of interest.
Using the dynamic approach, Davis manipulates the linearized hydrodynamic
equations into a balance equation for the ‘disturbance energy’, from which static
stability results are obtained. For example, the boundaries of the instability window for the rivulet with pinned contact-lines is a function of scaled axial wavenumber α and given by
αc2 = 1 −
π2
.
(2π − θs )2
(4.1)
The limiting case of θs = 0 corresponds to a cylindrical interface touching the plane
along a generator line and yields an upper boundary of αc = (3/4)1/2 (note that the
Plateau limit αc = 1 cannot be recovered from (4.1) because of differences in the
class of disturbances admitted). In contrast, θs > π is a lens-like cylindrical interface resting on a parallel plane; this case is seen to be stable to all lengths. Davis’
stability limits recover the constant-volume limits of Brown & Scriven (1980) and
Majumbar & Michael (1976), as expected. In the present study, the disturbance
energy equations posed by Davis are solved in the case of an inviscid fluid to obtain
explicit dispersion relations as well as eigenmode structures for both natural and
pinned axial end-plane conditions.
87
Studies of stability to finite-amplitude disturbances must account for nonlinear effects and, for free boundary problems, the further complication of changes
in connectivity. Since any free surface can be plucked to a shape that is nearly
pinched off, there are always finite-amplitude disturbances that do not return to the
base-state. Restrictions on the size and type of disturbances for finite-amplitude
stability results are then to be expected. The characterization of such disturbances
depends on the Steiner symmetrization procedure. Whenever the symmetrization
is possible, it delivers static stability to a class of finite-size disturbances. When
symmetrization is not possible, no conclusion can be made. The problem considered is defined by the radius of the cylinder and its axial extent, the polar extent
of the cylindrical-cup constraint, and the fill-ratio of the liquid. A comprehensive
treatment would study how results depend on fill-ratio. To keep the focus on ideas
and main results, the role of fill-ratio has been suppressed. For the dynamics, fullcylinder volumes (fill-ratio of unity) are considered, while for the torus results the
role of volume will be addressed when it appears. For the symmetrization results,
volumes such that shape meets the support with common tangent are assumed,
although generalization to other volumes clearly would be straightforward.
In the sections that follow, the governing equations are formulated and reduced
to linear-operator form, from which a solution is constructed using a Rayleigh-Ritz
procedure on a constrained function space. Growth rates and eigenmode structures
are then reported. Second, the cylinder is lifted to the torus and the focus is on
wire constraints. Unlike the perfect cylinder, the location of the wire constraint
affects the stability. Next, symmetrization is used to establish a finite-amplitude
stability result for lens-like cylindrical interfaces. Finally, some concluding remarks
are offered.
88
(a)
(c)
(b)
r = RH1 + ¶ ΗHΘ,z,tLL
z
L
Figure 4.1: Constrained cylindrical interface definition sketch: (a) crosssection with solid support (thick) (b) axial-section with interface
pinned at endpoints and (c) 3D view, with sample disturbed interface.
4.2
Formulation of dynamical problem
A cylindrical interface of radius R and axial length L is constrained by a cylindricalcup solid support, of polar extent 0 ≤ θ ≤ θs in cylindrical coordinates (r, θ, z), as
shown in figure 4.1. The domain of the liquid bridge,
D ≡ {(r, θ, z) | 0 < r ≤ R, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ L},
89
(4.2)
is bounded by a free surface (4.3a), the cylindrical-cup support (4.3b) and two
solid, parallel end-planes (4.3c);
∂Df ≡ {(r, θ, z) | r = R, θs ≤ θ ≤ 2π, 0 ≤ z ≤ L},
(4.3a)
∂D1s ≡ {(r, θ, z) | r = R, 0 ≤ θ ≤ θs , 0 ≤ z ≤ L},
(4.3b)
∂D2s ≡ {(r, θ, z) | r = R, 0 ≤ θ ≤ 2π, z = 0, L},
(4.3c)
∂Dint ≡ ∂Df ∪ ∂D1s .
(4.3d)
Additionally, it is also instructive to define the interface as the union of the free
surface and cylindrical-cup support (4.3d). The fluid of immersion has no inertia
and applies a constant pressure on the interface, while the effect of gravity is
assumed to be negligible.
The interface of this incompressible, inviscid fluid is given a small timedependent disturbance of the form η(θ, z, t). Two classes of end-plane boundary
conditions (4.3c) are considered. The first is the ‘natural’ boundary condition,
equivalent to an infinitely-periodic interface, for which the contact-line is free to
move with contact-angle π/2,
¯
∂η ¯¯
= 0,
∂z ¯z=0,L
(4.4)
while the second is a pinned contact-line
¯
η ¯z=0,L = 0.
(4.5)
Furthermore, the linearized interface disturbance is constrained by the integral
form of the incompressibility condition, which requires the disturbance to be volume conserving
Z
L
Z
2π
η (θ, z) dθ dz = 0.
0
0
90
(4.6)
4.2.1
Hydrodynamic equations
The flow is assumed to be irrotational and incompressible. Therefore, the velocity
field is described by a velocity potential, v = −∇Ψ, which satisfies Laplace’s
equation on the domain,
1 ∂
∇ Ψ(r, θ, z, t) =
r ∂r
µ
2
∂Ψ
r
∂r
¶
+
1 ∂ 2Ψ ∂ 2Ψ
+
= 0 [D] .
r2 ∂θ2
∂z 2
(4.7)
The kinematic condition
vr = −
∂Ψ
∂η £ f ¤
=−
∂D
∂r
∂t
(4.8)
relates the radial component of the velocity to the free surface deflection there.
Similarly, the no-penetration condition on the surfaces of support require
∂Ψ
= 0 [∂D1s ] ,
∂r
∂Ψ
vz = −
= 0 [∂D2s ] .
∂z
vr = −
(4.9a)
(4.9b)
The pressure field in the fluid domain is determined from the linearized
Bernoulli equation
p = p0 + ρ
∂Ψ
∂t
[D] ,
(4.10)
where ρ is the fluid density and p0 is the static pressure required to maintain the
fluid’s static interface shape. Deviations from equilibrium surface configurations
generate pressure gradients, and equivalently capillary-driven flows, described by
the Young-Laplace equation,
p/σ = 2H ≡ κ1 + κ2
£
¤
∂Df ,
(4.11)
which relates the principal curvatures, κ1 and κ2 , of an interface held by surface
tension σ to the pressure there. Accordingly, a constant pressure is the criteria for
static equilibrium, such as p0 = σ/R for a cylinder.
91
4.2.2
Normal-mode reduction
The motion of an inviscid fluid, whose cylindrical interface is held by surface tension
is governed by the linearized field equations (4.6)-(4.11), which can be reduced to
an eigenvalue problem through the use of normal modes. To begin, normal modes
Ψ (r, θ, z, t) = φ (r, θ, z) eγt , η (θ, z, t) = y (θ, z) eγt
are substituted into (4.7)-(4.9) to produce a boundary value problem for
µ
¶
1 ∂
∂φ
1 ∂ 2φ ∂ 2φ
2
∇ φ(r, θ, z) =
r
+ 2 2 + 2 = 0 [D] ,
r ∂r
∂r
r ∂θ
∂z
£
¤
∂φ
= −γy(θ, z) ∂Df ,
vr = −
∂r
∂φ
vr = −
= 0 [∂D1s ] ,
∂r
∂φ
vz = −
= 0 [∂D2s ] .
∂z
(4.12)
φ
(4.13a)
(4.13b)
(4.13c)
(4.13d)
To solve (4.13), consider boundary conditions (4.13b) and (4.13c) as a single boundary condition on the interface (4.3d) and restrict the ‘interface’ disturbances to
have no amplitude on the surface-of-support (4.3b). Such constrained interface
disturbances satisfy the no-penetration condition (4.13c) by construction and the
solution to (4.13) is given by
φ(r, θ, z) = −
∞ X
∞
X
m=1 l=0
where
ξl ≡
¡
¢
³ mπ ´
r
2 Il mπ
L
¢
¡
γ 2
cos
z ξl [Blm cos (lθ) + Clm sin (lθ)],
π m Il0 mπ
L
R
L
(4.14)



1/2 l = 0


1
l 6= 0,
Z L Z 2π
³ mπ ´
z cos (lθ) dθ dz,
Blm ≡
y (θ, z) cos
L
0
0
Z L Z 2π
³ mπ ´
z sin (lθ) dθ dz,
Clm ≡
y (θ, z) cos
L
0
0
92
(4.15a)
(4.15b)
(4.15c)
and Il is the modified Bessel function of the first kind.
Given the velocity potential solution (4.14) and the normal mode reduction
of (4.10,4.11), one may equate the capillary pressure from (4.11) to the inertial
pressure from (4.10), evaluated at the interface (4.3d), to give
i
h y
yθθ
γ − 2 − 2 − yzz = ρi γφ|r=R ,
R
R
(4.16)
which is recognized as an integro-differential equation governing the motion of the
interface.
4.2.3
Reduction to operator equation
The integro-differential equation (4.16) is reduced to a problem on the polar component of the interface disturbance y(θ, z) for natural (4.4), and pinned (4.5) end
conditions.
Natural wetting condition on endplate
To enforce the natural endpoint condition (4.4), a series solution
y (θ, z) =
³ πz ´
Ak (θ) cos k
,
L
k=1
∞
X
(4.17)
is used to reduce (4.16) to an integro-differential eigenvalue problem on Ak (θ),
∞
¡
¢
1 X
00
Ak + 1 − α2 Ak = λ2
Il (α) ξl [Bl cos (lθ) + Cl sin (lθ)],
απ l=0
93
(4.18)
with
λ2 ≡ ργ 2 R3 /σ,
(4.19a)
α ≡ kπR/L,
(4.19b)
0
Il (α) ≡ Il (α) /Il (α) ,
Z 2π
Z
Bl ≡
Ak (θ) cos (lθ) dθ, Cl ≡
0
(4.19c)
2π
Ak (θ) sin (lθ) dθ.
(4.19d)
0
Equation (4.18) is recast as an eigenvalue problem on linear operators
K n [Ak ; α] = λ2 M n [Ak ; α] ,
¡
¢
00
K n [Ak ; α] ≡ Ak + 1 − α2 Ak ,
∞
1 X
n
M [Ak ; α] ≡
Il (α) ξl [Bl cos(lθ) + Cl sin(lθ)].
απ l=0
(4.20a)
(4.20b)
(4.20c)
Pinned condition on endplate
An analogous operator equation is derived for the pinned endpoint condition (4.5)
using a solution series of the form;
µ
¶
2πz
Ak (θ) sin k
y (θ, z) =
.
L
k=1
∞
X
(4.21)
Substitution of (4.21) into (4.16) produces an integro-differential equation on
Ak (θ), parameterized by axial wave-number k, aspect ratio α̂ ≡ πR/L and formulated as an eigenvalue problem
K p [Ak ; α̂, k] = λ2 M p [Ak ; α̂, k] ,
¡
¢
00
K p [Ak ; α̂, k] ≡ Ak + 1 − (2k α̂)2 Ak ,
¶
X 64 µ
k2
p
×
M [Ak ; α̂, k] ≡
π 3 mα̂ 4k 2 − m2
m−odd
X
Il (mα̂) ξl [Bl cos (lθ) + Cl sin (lθ)].
(4.22a)
(4.22b)
(4.22c)
l
Unlike (4.20a), the pinned operator equation (4.22a) is parameterized by the aspect
ratio α̂ and discrete axial wave-number k.
94
No-penetration auxiliary condition
Solutions of the natural/pinned operator equations are required to satisfy the incompressibility (4.6) and no-penetration (4.13c) conditions. The solution series
for the natural (4.17) and pinned (4.21) endpoint conditions were chosen to satisfy the incompressibility condition (4.6). However, to ensure the no-penetration
condition (4.13c), equations (4.20a) and (4.22a) are augmented with the following
restriction,
Ak (θ) = 0, 0 ≤ θ ≤ θs .
(4.23)
This auxiliary condition (4.23) is satisfied by restricting candidate solutions of
(4.20a) and (4.22a) to an appropriately chosen function space.
4.2.4
Solution of operator equations
The Rayleigh-Ritz procedure for linear operators (Segel, 1987) is used to solve
the natural (4.20a) and pinned (4.22a) operator equations, which have the same
structural form. The necessary input to such a procedure is a set of orthonormal
basis functions, which span a predetermined function space.
Constrained function space
To construct the constrained function space, begin by considering candidate functions of the form
Ak (θ) =



0
0 ≤ θ ≤ θs


h(θ) θs ≤ θ ≤ 2π.
95
(4.24)
By definition, functions of the form (4.24) satisfy (4.23). To ensure the interface
perturbation Ak (θ) is single-valued, the following conditions are placed on the free
surface deformation h(θ),
h(θs ) = 0, h(2π) = 0.
(4.25)
Next, assume the free surface deformation y(θ) may be expressed as
h(θ) =
N
X
bj cos(jθ) + cj sin(jθ).
(4.26)
j=0
Substituting (4.26) into (4.25) gives




 1 cos(θs ) · · · cos(N θs ) sin(θs ) · · · sin(N θs )  




1
1
···
1
0
···
0

b0
..
.
cN
 

  0 
 =   , (4.27)


0
or a set of algebraic equations on the coefficients of the test function series (4.26).
There exists 2N − 1 coefficient vectors that solve (4.27), or 2N − 1 linearly independent functions of the form (4.26) that solve (4.25). The Gram-Schmidt orthogonalization procedure and a computer algebra package are used to convert the
linearly independent basis functions into a set of orthonormal functions {ψj (θ)},
which span the constrained function space. For reference, a sample test function
ψ is shown in figure 4.2.
Rayleigh-Ritz method
Finally, a solution series
Ak (θ) =
2N
−1
X
j=1
96
aj ψj (θ),
(4.28)
Ψ
Θs
2Π
Π
Θ
Figure 4.2: Sample orthonormal basis function ψ(θ)
constructed from the orthonormal basis functions ψj is used to reduce the eigenvalue operator equation of form K [Ak ] = λ2 M [Ak ] to a matrix equation,
Kij aj = λ2 Mij aj ,
Z
Z 2π
K [ψi ] ψj dθ; Mij ≡
Kij ≡
(4.29a)
2π
M [ψi ] ψj dθ.
(4.29b)
0
0
The eigenvalues and eigenvectors of (4.29a) are readily computed using standard
numerical routines. Eigenvalues and eigenvectors are specified by integer pairs
[l, k], the polar and axial wave-numbers, and both depend on the continuous aspect
ratio πR/L. The polar wave-number is simply the number of intersections plus
one of the disturbed circle with undisturbed circle (c.f. Figure 4.7). Given an
(r)
eigenvector aj of (4.29a), the corresponding eigenfunction is
(r)
Ak (θ)
=
2N
−1
X
j=1
97
(r)
aj ψj (θ).
(4.30)
(a)
(b)
Λ2
Λ2
0.25
0.05
k‡1
k‡2
k‡3
0.20
0.15
0.03
0.10
0.02
0.05
0.00
0
k‡1
k‡2
k‡3
0.04
0.01
L
10
20
30
40
0.00
0
50 R
L
20
40
60
80
100 R
Figure 4.3: Growth rate of most unstable polar mode [1, k] vs. aspect ratio
L/R (a) for θs = 0.01 with natural conditions and (b) for θs = 2.0
for pinned conditions.
4.3
Results for a constrained cylindrical interface
The eigenvalues/eigenmodes of the operator equations (4.20a) and (4.22a) are computed using standard numerical routines for a fixed constraint size θs . All eigenvalues presented here show convergence to within 0.1%, when 13 terms (N = 7)
have been used in the solution series (4.28).
The eigenvalues for the most unstable (λ2 > 0) polar mode (l = 1) depend on
the aspect ratio as shown in figure 4.3, for various axial wave-numbers k. As seen
from the natural operator (4.20a), L/R and k are not independent and the separate
curves collapse onto a single curve by scaling. In contrast, because L/R and k
appear separately in the pinned operator (4.22a), they appear to be independent
but the same scaling also collapses the curves in figure 4.3(b) so, in fact, they are
not independent.
Relative to the natural end-plane constraint, the pinned end-plane constraint
always increases static stability αc by a factor of two, which is shown for the
98
(a)
(b)
Λ2
0.4
0.3
Λ2
0.4
Natural
0.3
Pinned
0.2
0.2
0.1
0.0
Rayleigh
Θs =0.01
Θs =2.0
Θs =2.5
0.1
0.2
0.4
0.6
0.8
Α
0.0
1.0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
0.2
0.4
0.6
0.8
Α
1.0
Figure 4.4: Dispersion relation of l = 1 modes (a) for θs = 0.01 with natural
and pinned conditions and (b) for varying θs for natural conditions. Rayleigh dispersion, for reference.
touching wire in figure 4.4(a). Increasing the extent of polar constraint also always stabilizes, as shown for the natural conditions in figure 4.4(b). These static
stability results are summarized in figure 4.5. The curve for natural end conditions on figure 4.5(a) recovers equation (4.1) and could be read off figure 4.4(b).
The pinned curve on figure 4.5(a) is the natural curve scaled by a factor of two.
This relationship between the pinned and natural stability limits is seen again in
figure 4.5(b), which plots the static limit as it depends on the length and polar
constraints, the so-called stability envelope. There, the k = 1 pinned and k = 2
natural curves collapse to a single curve and the k = 1 pinned and k = 1 natural
curves differ by a factor of two. Indeed, the four curves of figure 4.5(b) all collapse onto a single curve. They have plotted separately to compare directly to the
natural end-constraint results of Brown & Scriven (1980), with which they are in
agreement. The convergence of eigenvalue/eigenmodes and the recovery of a range
of static stability results are tests that validate the computations.
Each dispersion relation seen in figure 4.4 exhibits a fastest growing mode with
corresponding aspect ratio which, following Rayleigh, can be interpreted as an es-
99
(a)
(b)
L
Αc
0.8
Natural
R
35
k=1 Natural
Pinned
30
k=2 Natural
0.6
k=1 Pinned
25
k=2 Pinned
20
0.4
15
10
0.2
5
Π
2
Π
Θs
0
Π
2
Π
Θs
Figure 4.5: Static stability against polar constraint θs measured by (a) wavenumber αc or (b) envelope of stable L/R (below curve).
(a)
(b)
Λ2m
Αm
0.25
0.6
Natural
Pinned
0.20
Natural
0.5
Pinned
0.4
0.15
0.3
0.10
0.2
0.05
0.1
Π
2
Π
Θs
Π
2
Π
Θs
Figure 4.6: (a) Growth rate λ2m and (b) wave-number αm against polar constraint for fastest growing mode.
timate for the bead size which results from the breakup instability. Figure 4.4(b)
shows that the maximum growth rates of the unstable modes are smaller than that
for the corresponding unconstrained Rayleigh jet (λ2Ray = 0.343). The Rayleigh
dispersion relation is plotted for reference. Figure 4.6 summarizes the maximum
growth rates (fig. 4.6(a)) and corresponding aspect ratios (fig. 4.6(b)) as they depend on extent-of-support. Figure 4.7(a) illustrates the mode shape of a pinnedend disturbance, unstable when unconstrained, but stabilized by a cup that is
63.7% of the critical cup (π). For the same constraint, figure 4.7(b) shows the
100
(a)
(b)
(c)
(d)
Figure 4.7: Modes [l, k] in 3D and polar projection for L/R = 2π with (a, b)
θs = 2.0 and pinned conditions for (a) [1,1] and (b) [3,2] and (c, d)
θs = 0.01 and natural conditions for (c) [1,1] and (d) [2,3].
stable [3, 2] vibrational mode. Increasing the polar constraint stabilizes, as mentioned. However, slower growing unstable modes remain and such a mode shape
is illustrated in figure 4.7(c) for natural-end conditions in the presence of a wire
constraint. For reference, figure 4.7(d) shows the [2, 3] vibrational mode (stable)
for a wire constraint.
Explicit knowledge of the inertia operator M [y] is needed to determine the
spectrum of the eigenvalue problem (4.29a), but is not necessary to show static
stability, provided one can show (M [y; α], y) > 0 for y 6= 0. Indeed, if M [y; α] is
positive definite, the curvature operator K[y; α] determines the sign of the eigenvalues and may be used to determine the static stability limit αc . Positive-definiteness
101
of M is straightforward to show and the argument is sketched. Applying Green’s
first identity to the velocity potential φ(x) on the domain D, results in an integral
over the domain and its boundary,
Z
Z
£
φ (∇φ · n) dS =
∂D
¤
φ∇2 φ + ∇φ · ∇φ dV .
(4.31)
D
The domain integral is evaluated using ∇2 φ = 0. The boundary integral is evaluated using φ|∂D = M [y], which follows from the linearized Bernoulli equation, and
using ∇φ · n = y, which follows from the kinematic condition.
For the axial and polar constraints considered so far, narrowing the class of
disturbances enhances stability. Next, the class of equilibrium states is enlarged
to show that, relative to the cylindrical cap interface, constraint can be stabilizing
or destabilizing.
4.4
Lifting the cylinder to the torus
The torus with surface-of-revolution constraint is of interest, c.f. figure 4.8. The
torus is described using a standard parametric representation (Kreyszig, 1991),
x = (R + a cos θ) cos ϕ, y = (R + a cos θ) sin ϕ, z = a sin θ.
(4.32)
The torus is a near-cylinder for ² ≡ a/R ¿ 1 and, in the limit ² → 0, a cylinder
with axial periodicity. An in-plane bending of the constraint introduces a secondary
curvature, controlled by ². A straightforward calculation of the mean curvature H
of the torus (4.32) shows that it varies with θ according to a(2H) = 1+²(cos θ/(1+
² cos θ)). A non-constant mean curvature does not satisfy the Young-Laplace law
and, hence, the torus surface is not in capillary equilibrium. However, for the nearcylindrical torus, a near-toroidal shape can be found that is an equilibrium shape
102
(a)
(b)
z
Θs
R
a
Θ
xy
Figure 4.8: Torus sketch in (a) 3D view and in (b) polar view with cup support (thick line).
provided that the shape is constrained. It turns out that such a constraint must
be symmetric about the z = 0 mid-plane. For this reason, constraints symmetric
about the mid-plane and subtending an angle θs are considered. The constraints in
focus are directly analogous to those considered for the cylinder. This constraint is
either ‘inside’ (as in figures 4.8 and 4.10(b)) or ‘outside’ (as in figure 4.9(b)) relative
to the origin. In the limiting case θs = 0, the constraint is a wire. In the further
limit of ² = 0, the perfect cylinder is recovered and the distinction between inside
and outside is lost.
The family of near-toroidal equilibrium shapes is referred to as the ‘torus-lift
of the cylinder’; the family is parameterized by ². The torus-lift is of interest for
two different kinds of results, both regarding static stability. First, the stability of
the torus-lift to small disturbances for the case of inner and outer wire constraints
is calculated. This suffices to demonstrate the destabilizing and stabilizing nature
of the inner and outer constraints, respectively. Second, Steiner symmetrization
on the torus-lift is applied to cylindrical lens-like shapes (θs > π) to demonstrate
stability to finite-amplitude disturbances. The limiting case θs = π is seen to be a
‘hard’ stability boundary.
103
z
ra=1+Ε r0 HΘL
Θ
xy
Figure 4.9: Torus equilibrium shape r0 (θ) with unit-circle (dotted) for reference.
4.4.1
Near-toroidal equilibrium shapes
Lengths can be scaled by a so small deformations take the form (c.f. figure 4.8),
r/a = 1 + ²r0 (θ).
(4.33)
Here, in view of the mean curvature of the perfect torus, shapes depending only on
θ are anticipated. The strategy is to seek near-toroidal shapes that just cancel the
curvature of the torus. At order ², the near-cylindrical and near-toroidal curvatures
00
are found to be cos θ/a and −(r0 + r0 )/a, respectively. Hence, to this order,
equilibrium is maintained provided,
00
r0 + r0 = cos θ.
(4.34)
r0 (θ) = 1/2 (θ − π) sin θ,
(4.35)
Solution to (4.34) is given by
where boundary conditions symmetric about the mid-plane corresponding to the
outer wire constraint have been enforced. This particular shape, which has a corner
at the wire (θ = 0, 2π), is shown in figure 4.9. The corner may be shifted from the
outer to the inner position by redefining the polar surface coordinate θ → θ − π
and using an alternative solution to (4.34), r0 = 1/2(θ − π) sin(θ − π).
104
Note that a slightly larger class of solutions would be obtained by adding a constant to the right-hand-side of (4.34). This corresponds to allowing near-toroidal
equilibrium shapes with ‘constant’ mean-curvatures, of a different ‘constant’ than
the cylinder. Physically, this amounts to an adjustment of the volume of liquid.
In summary, if one demands that the volume of the equilibrium shape be the same
as that of the corresponding near-cylindrical shape, it can be arranged, although
there is no fundamental reason to do so. Finally, volumes can be easily calculated
by Pappus’ theorem knowing the centroid of the plane figure and its planar area.
4.4.2
Stability of near-toroidal equilibrium shapes
The equilibrium shape is perturbed by an amount δ of the form r1 (θ, ϕ)
r/a = 1 + ²r0 + δr1 (θ) eiqϕ ,
(4.36)
where q is the toroidal azimuthal wave-number. The disturbance must remain
smaller than the deviation from the cylinder, δ ¿ ². Note that the disturbance
preserves volume of the equilibrium state.
Static stability of the base-state is determined by the order δ deviation of the
mean curvature. This deviation, as it depends on ², is formulated as an operator
£
¤
K[r1 ; α, ²] ≡ (2H)δ = (1 + ² cos θ)2 + ²2 cos θ − α2 r1
(4.37)
0
2 00
2
2
∓² sin θ (1 + ² cos θ) r1 + (1 + ² cos θ) r1 − ² (1 + ² cos θ) F [θ] ,
where α = qa/R is the scaled wave-number and
F [θ] ≡
¡
¢
¤
1 £
−3 − (θ − π)2 + (θ − π)2 − 13 cos 2θ + 14 (θ − π) sin 2θ .
16
(4.38)
Allowable solutions of (4.37) are subject to the no-penetration condition along the
wire constraint,
r1 (θ = 0, 2π) = 0.
105
(4.39)
(a)
(b)
Αc
z
Εc
1
ra=1+Ε r0 HΘL
0.5
Θ
Inner wire
Outer wire
Cylinder wire
0
0
0.1
0.2
0.26
Ε
xy
ra=1+Ε r0 HΘL+∆ r1 HΘ,jL
Figure 4.10: Stability of toroidal equilibrium shapes showing (a) static limit
αc against deviation from cylinder ² and (b) typical destabilizing
mode shape, with equilibrium shape (dotted) for reference.
The ∓ signs in (4.37) denote a torus with an outer or inner wire constraint, respectively.
Static stability is determined by the sign of the curvature operator (K[y; α], y),
where the surface disturbance y belongs to an appropriate function space. Allowable surface disturbances to the torus must satisfy (4.39), which are structurally
identical to the free surface disturbances of the cylinder with a wire constraint.
Therefore, the solution series (4.28) derived earlier can be used as the input to
the Rayleigh-Ritz procedure on the curvature operator (4.37) for the torus. Also
note that, the curvature operator (4.37) reduces to the curvature operator for the
cylinder (4.20b) in the limit ² → 0.
Given α and ², the spectrum of K[r1 ; α, ²] = λ2 r1 is readily computed using
the Rayleigh-Ritz procedure. For secondary curvature ², the critical wave-number
αc is computed by iterating over α and locating where the unstable eigenvalue
changes sign.
Figure 4.10(a) plots the critical wave-number vs. secondary curvature for a
torus with a wire constraint placed on the inner and outer extrema, while fig106
(a)
(b)
AA' Section
BB' Section
Figure 4.11: Symmetrization of general shape proceeds from (a) non-circular
slice AA0 to (b) circular slice BB0 with reassembly to axisymmetric shape.
ure 4.10(b) displays the polar projection of the instability mode shape. As shown
in figure 4.10(a), the location of the wire constraint can have either a stabilizing (outer pin) or destabilizing (inner pin) effect, when compared to the wireconstrained cylinder. For the inner pin, as the secondary curvature increases, a
cancelation occurs at ²c = 0.26, where the destabilizing effect of the secondary
curvature exactly cancels the stabilization of the wire inherited from the cylinder.
The toroidal equilibrium shape of this size with inner wire constraint will have
precisely the Plateau-Rayleigh limit. The validity of the expansion to ²c and beyond is plausible in view of the linear dependence seen in figure 4.10(a) but has
not been investigated further.
107
4.4.3
Symmetrization and large-amplitude stability of the
lens meniscus
To illustrate symmetrization and its relationship to stability, consider a liquid
bridge spanning end-plates where it is pinned along circular contact-lines, figure 4.11. Symmetrization starts with the non-symmetric shape, figure 4.11(a)
and ends with the axisymmetric shape, figure 4.11(b). First, take a ‘constant z’
cut to obtain the plane figure of section AA0 . This AA0 figure encloses a planar
area. The perimeter of this figure is then minimized, for fixed planar area, to
obtain a circle, section BB0 . This circle is then reassembled with the circles from
other sections, aligning centers, to yield the axisymmetric shape, figure 4.11(b).
Symmetrization establishes a mapping from non-axisymmetric to axisymmetric
shapes. Gillette & Dyson (1972) proved that this mapping preserves volume while
decreasing surface-area. One should note that their proof doesn’t preclude shape
differences of finite amplitude. They applied their result to small non-axisymmetric
disturbances to conclude that for axisymmetric bridge base-states, single-valued in
radial coordinate, it is sufficient to consider axisymmetric disturbances when testing for stability. That is, if an equilibrium configuration is statically unstable to a
non-axisymmetric disturbance, then it is unstable to an axisymmetric disturbance.
Note that the procedure fails if the disturbance section AA0 is not simplyconnected. Such sections certainly occur for base-states that are multiple-valued
in radius, as for the dotted shape in figure 4.11(b) with slice taken above the top
plate. Shapes with tangents to the endplates are the limiting shapes, known as the
‘rotund limit’ (Russo & Steen, 1986). Separate analysis assuming small amplitudes
have shown that, beyond the rotund limit, non-axisymmetric disturbances are
in fact destabilizing (Slobozhanin et al., 1997). Hence, failure of the procedure
108
corresponds to a sharp bound. Finally, note that the procedure succeeds even
with a solid internal boundary, such as a rod spanning the endplates, as long as
the base-state and disturbances maintain a fully-wetted internal boundary. This
occurs, of course, since it is increases/decreases in the liquid/gas surface area only
that alter the energy. To use the Gillette & Dyson result for cylindrical interfaces,
the cylinder is lifted to the torus.
Symmetrization of the liquid bridge uses a cylindrical coordinate system and
takes plane slices orthogonal to the generator axis, which shall be denoted the
symmetrization-axis, (c.f. figure 4.1). Symmetrization can proceed if the slices
yield a simply-connected planar figure. This occurs precisely when starting with
a single-valued volume-enclosing shape.
This motivates the notion of finite-
amplitude ‘admissible’ interfaces – interfaces that are single-valued in the radial
direction relative to the symmetrization-axis. To obtain stability results, both
base-state and disturbances must be admissible.
The equilibrium interfaces of the torus-lift family are admissible for θs > π and
a range of volumes. These are lens-like meniscus shapes. Recall the Gillette &
Dyson result: symmetrization maps non-axisymmetric shapes onto axisymmetric
shapes of the same volume but with lesser surface area (GD reduction theorem).
This applies directly to the torus-lift. Hence, to obtain stability to general volumepreserving admissible disturbances, it is sufficient to consider axisymmetric disturbances, relative to the symmetrization axis. These are the most dangerous. This
reduces the full 3D problem to a 2D planar one. Of course, the reduction fails for
multiple-valued base-states; these do occur for θs < π and a filling ratio of unity. It
should be noted that volume, an independent parameter, is taken to be the filling
volume here. The filling-volume is that volume whose equilibrium shape meets the
109
constraint at the contact-line tangentially. Clearly, for a fixed θs < π, there will
be a range of volumes (low) that have single-valued and a range (modest to high)
that have double-valued equilibrium shapes.
Symmetrization of the torus-lift can deliver general large-amplitude stability
results in the limit of the cylindrical-lens meniscus, figure 4.12. That is, in the
cylindrical limit ² = 0, the toroidal equilibrium shapes reduce to cylindrical cap
shapes and, for θs > π, these are cylindrical lenses. The GD reduction theorem
applies to each ² > 0 and thus holds in the limit ² = 0. The toroidal equilibrium
shape is relatively stable to axisymmetric disturbances (finite-amplitude, admissible, volume-preserving) and, hence, the cylindrical lens is relatively stable to
disturbances varying in parallel cross-sections, figure 4.12(a). It is left to show
that the lens is also relatively stable to disturbances in the transverse plane. his
follows by choosing a new symmetrization axis – the axis of the cylindrical meniscus – for which slices are transverse cross-sections, figure 4.12(a), and applying the
GD theorem again. One then concludes that disturbances with variations along
parallel cross-sections are most dangerous. Together, the results from orthogonal symmetrization procedures, delivers the result; the cylindrical lens is stable to
general large-amplitude volume-preserving admissible disturbances. It remains to
state this result formally.
Disturbance class.
‘Admissible’ interface shapes are those with parallel and
transverse cross-sections that are simply-connected. (An equivalent definition is
those shapes that are single-valued in the radial coordinate, relative to both symmetrization axes; that is, both parallel and transverse slices give single-valued
shapes.)
Figure 4.12(b) illustrates a large-amplitude disturbance (transverse section)
110
(b)
(a)
(c)
(d)
Figure 4.12: Symmetrization of lens meniscus using (a) parallel and transverse cross-sectional slices in 3D view; sketches of largeamplitude disturbances that are (b) admissible (in transverse
section), or (c, d) inadmissible (c) in transverse section, (d) in
parallel section.
that is admissible while figures 4.12(c, d) give orthogonal views of an inadmissible
disturbance.
Theorem 1 The cylindrical interface of cylindrical volume, constrained by solid
support θs > π with pinned contact-lines, is statically stable to arbitrarily large
volume-preserving disturbances as long as the disturbances are admissible and preserve wetted area-of-contact with the solid support.
111
This theorem is proved by observing that i) the GD reduction theorem applies to large-amplitude disturbances, ii) the GD reduction applies to the lens-like
toroidal equilibrium states (inside constraint), iii) the toroidal azimuthal coordinate becomes the cylindrical axial coordinate in the cylindrical limit of the torus,
and that iv) the GD reduction also applies with the toroidal axial coordinate as
symmetrization axis. In summary, the cylindrical lens base-state admits symmetrization in two orthogonal directions, by which it is found to be globally stable
to admissible disturbances. Theorem 1 says that θs > π is sufficient for stability
of the cylindrical interface while the linear stability results in the literature show
that θs ≥ π is also necessary for stability since, for θs < π, small disturbances that
destabilize have been exhibited.
4.5
Concluding remarks
Constraint tends to stabilize, whether via end-plane (axial) or lateral-cup (polar).
To the extent that a constraint narrows the class of disturbances to which a basestate is subject, this stands as a principle. In going from the natural to the pinned
end-plane conditions, the instability window is narrowed by a factor of two no
matter the extent of the cup constraint. Likewise, in going from the unconstrained
cylinder (Plateau-Rayleigh) to the cylinder touching a wire, the instability window
is narrowed by 13%. Increasing θs narrows the instability window further until it
disappears at θs = π and beyond which it remains closed. In contrast, bending the
cup constraint into a torus configuration can either stabilize or destabilize depending on whether the cup is placed on the inside or outside of the bend. Bending
creates two new families of equilibrium states. For each member of these families,
increasing the constraint will enhance stability, while relative to the constrained
112
cylinder the new family can be more or less stable. It has been shown how the
destabilization of bending curvature can cancel the stabilization of the wire to yield
the same critical value as occurs for the Plateau-Rayleigh instability. All these are
small-amplitude static results.
The torus-lift of the cylinder allows one to prove a large-amplitude stability
result for cylindrical lens interfaces (π < θs < 2π). Large disturbances, just so
they are single-valued, are statically stable. This may provide important insight
for application. For example, one way to discourage multiple-valued disturbances
is to limit the liquid volume. Furthermore, as a tool for obtaining finite-amplitude
results, the symmetrization technique may well be useful to a broader class of
interfacial stability problems.
As regards dynamics, the disturbance equations posed by Davis (1980) are
solved in the special case of an inviscid liquid when the liquid is pinned along
the edges of the circular-cup and of a volume that just fills the circular section.
An integro-differential equation has been derived for both natural and pinned
axial constraints. The governing equation is formulated as an eigenvalue problem
on linear operators, from which a solution is constructed using a Rayleigh-Ritz
procedure. The eigenvalues/eigenmodes are dependent upon the polar constraint
and have been computed from a truncated set of linear algebraic equations using
standard numerical routines.
The lens-like interfaces are always stable; only oscillatory motions occur. For
the drop-like interfaces, only the eigenmode with smallest polar wave-number l = 1
exhibits instability and that for a range of axial wave-numbers. The polar constraint stabilizes axial wave-numbers that are unstable for the Plateau-Rayleigh
cylinder, as mentioned. The constraint also slows the growth rate of the unsta-
113
ble modes but the slowing of growth rates does not scale in the same manner as
the static limit stabilization. For example, the disturbance with fastest growth
rate for the natural end-point conditions grows nearly three times as fast as that
of the corresponding pinned end-point mode. The cylinder-against-a-wire illustrates the influence of constraint. Compared to Rayleigh break-up, the fastest
growing disturbance is 18% longer suggesting about a 6% increase in final droplet
size. What might be easier to distinguish in experiment is that the characteristic
time to break-up is nearly 40% longer for the wire constraint as compared to the
unconstrained Rayleigh break-up.
114
CHAPTER 5
STABILITY OF THE SESSILE DROP: CONTACT-LINE
DYNAMICS AND SYMMETRY BREAKING
5.1
Introduction
The natural oscillations of the sessile drop are of fundamental interest in a number
of industrial applications, such as coating processes, ink-jet printing and spraycooling via drop-atomization. In contrast to the free drop, which has been studied
extensively since the time of Lord Rayleigh (1879), the existing literature on the
sessile drop is sparse and particularly so for analytical-based solution methods.
This occurs because supported drops possess less symmetry than free drops. Many
authors have extended the pioneering work of Lord Rayleigh, who considered the
linear oscillations of an incompressible, inviscid free drop. Lamb (1932) has used
the irrotational solution to compute the dissipation and equivalently the decay-rate
of free viscous drops. Much later, the viscous correction to the oscillation frequency
under the irrotational assumption was reported by Padrino et al. (2007). They
compare the spectrum of the free viscous drop computed using the irrotational
approximation to that using viscous potential flow theory (Joseph, 2006). In the
limit of small viscosities, the irrotational approximation is shown to be quite satisfactory and compares well with the exact solution of the linearized viscous problem
proposed by Miller & Scriven (1968) and computed by Prosperetti (1980b). The
aforementioned results have assumed the interface disturbances are small. With
regards to finite-amplitude disturbances, Tsamopoulos & Brown (1983) use a domain perturbation method to show that the characteristic frequencies of the free
drop decrease with the square of the amplitude in the moderate-amplitude regime.
115
Trinh & Wang (1982) have shown experimentally that a further increase in amplitude can lead to drop break-up. This has been verified numerically by Lundgren
& Mansour (1988), who use a boundary-integral method to identify the largeamplitude states that precede this type of break-up. In general, large amplitude
oscillations are handled using computational-based approaches, such as those developed by Patzek et al. (1991) for inviscid free drops.
Interest in the dynamics of supported drops, in contrast to free drops in zerogravity environments, can be attributed to a number of terrestrial applications.
One of the first works in the subject was the experimental study of Rodot et al.
(1979), who were concerned with characterizing the frequency and mode shape of
the supported drop as a function of the material parameters and support geometry.
This investigation was followed by the theoretical study of Strani & Sabetta (1984,
1988), who considered both inviscid and viscous drops under an idealized sphericalbowl constraint. Here the choice of support-surface greatly simplifies the analysis.
They use a Green’s function approach and a Legendre series expansion to reduce
the governing equation to a standard algebraic eigenvalue problem. Similarly,
Bauer & Chiba (2004, 2005) have considered the same problem but model the
spherical-bowl support as a large number of point-wise constraints in order to
compute the drop’s frequency spectrum. Ganan & Barerro (1990) use the Green’s
function expansion technique of Strani & Sabetta (1984) to analyze the linearized
dynamics of captive drops and bridges. Their numerical-based method is able to
handle gravitational effects and has been verified against experiment, which they
also perform. In contrast to isolated drops, supported drops require a model for
the three-phase contact-line. The aforementioned studies have made the common
assumption that the interface has an immobile or ‘pinned’ contact-line.
116
Α
Αr
Αa
uCL
Figure 5.1: Typical experimental relationship between contact-angle α and
contact-line speed uCL , which also shows the advancing αa and
receding αr static contact-angle (uCL → 0).
The dynamics of the moving contact-line are not as straightforward. One has
to establish the appropriate constitutive laws there (Dussan, 1979). As observed
experimentally, the typical relationship between the contact-angle α and contactline speed uCL is shown in figure 5.1. The ambiguity in the static contact-angle
(uCL → 0) is referred to as contact-angle hysteresis. More precisely, in many physical systems there exists a range of contact angles α ∈ [αr , αa ] where no contact-line
motion is observed. Here αa and αr are related to the material properties of the
advancing/receding contact-line, respectively. Figure 5.1 has served as the basis
for a number of theoretical studies on moving contact-lines and a few of the most
relevant studies will be mentioned here. In a series of studies on moving contactlines with application to fluid rivulets, Davis et al. assume that the contact-angle
depends smoothly on the contact-line speed, α = G(uCL ). They establish linear
stability bounds for a number of base-states (Davis, 1980; Weiland & Davis, 1981;
Young & Davis, 1987). Contact-angle hysteresis is excluded here because of assumptions on the smoothness of G. In a study of capillary-gravity waves of a
117
planar interface, Hocking (1987b) introduced




Θa (ξ − ξa ) ,




∂η
= 0,

∂t





Θr (ξ − ξr ) ,
the following contact-line condition
ξ > ξa
ξ r < ξ < ξa
(5.1)
ξ < ξr ,
which does allow for contact-angle hysteresis, despite the assumption that the
interface perturbation η is small. The ‘Hocking’ condition relates the linearized
contact-line speed ∂η/∂t to the deviation in contact-angle from its equilibrium
value ξ = ξ(η), which itself is a function of the interface disturbance η. Here Θa , ξa
and Θr , ξr are parameters related to the material properties; and the wetting and
spreading conditions of the advancing/receding contact lines, respectively. When
the deviation in contact-angle ξ exceeds the critical interval ξ ∈ [ξr , ξa ], contact-line
motion ensues. Hocking (1987a) has shown that in the absence of contact-angle
hysteresis ξa = ξr , (5.1) reduces to
∂η
= Θξ.
∂t
(5.2)
Similarly, the linearized version of the constitutive law adopted by Davis (α =
G(uCL )) can be written as
ξ=Λ
∂η
.
∂t
(5.3)
In this common limit, the contact-line conditions (5.2,5.3) are related through
the interpretation of the spreading parameters Θ = 1/Λ. For the purposes of
this chapter, the Davis interpretation (5.3) will be used in analyzing the dynamic
contact-line.
As regards the dynamics of drops in contact with a planar support, Lyubimov et al. (2004, 2006) study free and forced oscillations of the sessile drop with a
hemispherical base-state for both vertical and horizontal forcing. By exploiting the
118
symmetry of the base-state, they are able to compute the frequency spectrum under
a linearized Hocking boundary condition (5.2). Fayzrakhmanova & Straube (2009)
implement the full Hocking condition (5.1) via numerical integration to study the
stick-slip dynamics and frequency response caused by contact-angle hysteresis of
the forced hemispherical drop. In the absence of base-state symmetry, one generally turns to a more computationally-based method. Finite-element methods have
been employed by Basaran et al. to study the finite-amplitude natural oscillations
of pendant drops (Basaran & DePaoli, 1994), as well as the forced oscillations of
supported drops with application to drop ejection (Wilkes & Basaran, 2001) and
hysteretic response (Wilkes & Basaran, 1999), which was reported experimentally
by DePaoli et al. (1995). In another numerical study, James et al. (2003a) developed a Navier-Stokes solver to capture drop ejection phenomenon of forced sessile
drops.
Recently, there has been a growing interest in the dynamics of sessile drops
under external forcing with an emphasis on the wetting conditions at the threephase contact-line and their motion. In drop atomization experiments, James
et al. (2003b) observed a hierarchy of instabilities for the sessile drop forced by a
piezoelectrically-driven diaphragm. A more detailed investigation of these instabilities are given in Vukasinovic et al. (2007). A brief synopsis of the experimental
observations is given here.
To begin, a sessile drop with pinned contact-lines is placed on a vibrating
diaphragm and a frequency sweep is performed for fixed forcing-amplitude until resonance is observed or an axisymmetric interface disturbance is born. The
frequency is then locked and the forcing amplitude is increased. At a critical forcing amplitude, the contact-line de-pins and an azimuthal instability is triggered
119
along the contact-line. A further increase in the forcing amplitude results in the
azimuthal wave propagating upwards until it consumes the entire drop interface.
This time-dependent state is called the ‘lattice mode’. Finally, drop atomization is
observed at larger values of forcing amplitude. The azimuthal instability triggered
by the contact-line motion (de-pinning) will be discussed later.
In a similar study of mechanically-vibrated sessile drops, Noblin et al. (2004)
focused on the transition from a pinned to free contact-line and the acceleration
necessary to overcome the contact-angle hysteresis. Extending their previous work,
Noblin et al. (2005) showed that at sufficiently high accelerations an azimuthal
instability called the ‘triplon mode’ is generated in large fluid puddles. Once again,
this instability is triggered once the contact-angle hysteresis is overcome and the
contact-line de-pins. These modes and the instability observed in Vukasinovic et al.
(2007) exhibit subharmonic resonances, which are characteristic of Faraday waves
(Faraday, 1831). A thorough review of the parametrically-driven Faraday wave
instability is given by Miles & Henderson (1990).
As an alternative to vertical forcing, the sessile drop subject to in-plane (horizontal) forcing exhibits another set of intriguing phenomenon. The first being a
set of mode shapes that are odd or anti-symmetric about the vertical mid-plane.
These types of shapes are not possible under the assumption of axisymmetry, but
have been observed by Daniel et al. (2004) for sessile drops in forced oscillation on
chemically-treated gradient surfaces. In addition, they also observe translational
motion of drops that have overcome the hysteretic barrier or have mobile (unpinned) contact-lines. A study of both horizontal and vertical forcing by Noblin
et al. (2009) has shown that this translational motion can be directionally controlled. Surprisingly, Brunet et al. (2009) have shown that under external forcing
120
a sessile drop can overcome the influence of gravity and be driven up a sloped incline against gravity. A fundamental understanding of the wetting conditions that
give rise to these types of motions are of importance to micro-fluidic technology
and coating processes.
In this chapter, the small oscillations of the sessile drop under a number of
contact-line conditions are analyzed. Specifying the boundary conditions on the
three-phase contact-line sets the disturbance class to which the drop interface is
subject. The disturbances considered here fall into two categories and are termed
i) ‘kinematic’ or ii) ‘dynamic’. Kinematic disturbances have no ‘contact-line disturbance energy’ associated with their motions, while a dynamic disturbance can
dissipate energy. A kinematic disturbance either i) preserves the drop’s static
contact-angle (natural) or ii) has fixed contact-lines (pinned). The dynamic disturbance relates ‘field’ quantities by a postulated constitutive law valid on the
three-phase contact-line. More specifically, the dynamic contact-angle is related to
the contact-line speed via a spreading parameter.
A dynamic stability analysis is performed, whereby the linearized hydrodynamic equations, governing the interfacial motion of the inviscid sessile drop, are
formulated as a functional eigenvalue-problem on linear operators. The functional
equation is solved using one of two equivalent formulations, referenced as the ‘forward’ or ‘inverse’ problem. Both problems are reduced to a set of algebraic equations by the variational procedure of Rayleigh-Ritz. The frequency spectrum for
the kinematic disturbances, fixed contact-angle and fixed contact-line, are computed by solving the inverse problem. An ad hoc solution method is used for the
contact-line speed condition, which exploits the structure of the second variation
of surface energy and utilizes select results from the ‘natural’ disturbance class.
121
This problem is parameterized by azimuthal wave-number l, the static contactangle α (equivalently volume) and the boundary conditions by a spreading parameter Λ. Although the majority of the sessile drop motions are oscillatory, there does
exists an instability related to the natural disturbance for a range of contact angles
(90◦ < α < 180◦ ) that define the super-hemispherical base-state. The characteristic feature of this instability is an advancing contact-line, which is accelerating
relative to the receding contact-line, thereby transferring mass from one side of
the drop to the other. There is no such mechanism to resist this motion and the
droplet ‘walks’ along the supported surface. Cataloging this instability may be of
great importance in coating processes and microfluidic applications, where control
of the translational motion of fluid droplets is desirable.
The sessile drop with hemispherical base-state (α = 90◦ ) is a special case, since
it possesses some additional symmetry. It has characteristic oscillation frequencies that are degenerate with respect to azimuthal wave-number for the natural
disturbance class. To ‘break’ this symmetry, one can either i) change the static
contact-angle of the base-state, α 6= 90◦ or ii) introduce new boundary conditions Λ 6= 0. Pinning the drop’s contact lines lifts this degeneracy Λ → ∞.
There exists a range of frequencies for fixed polar wave-number k which depend
upon azimuthal wave-number. The frequency range is approximately 10% for
the pinned hemispherical base-state and can be increased significantly by varying the static contact-angle. Here, for a given polar wave-number k, the mode
shapes with largest azimuthal wave-number l have a greater oscillation frequency
in comparison to the low azimuthal wave-number shapes. This directional breaking
persists for super-hemispherical base-states (α > 90◦ ), where the high azimuthal
wave-number shapes have the larger oscillation frequencies. In contrast, for the
sub-hemispherical base-states (α < 90◦ ), the oscillation frequency decreases as the
122
azimuthal wave-number increases. The magnitude of the frequency range increases
as the volume deviates from hemispherical. In fact, for large deviations there is
the possibility for mode crossings, parameters at which two mode shapes share
the same frequency. An analogy can be made between the mode-crossing phenomenon and the ‘filling’ of the periodic table by energy levels. In addition to
modal crossings controlled by the base-state volume, they can be controlled by the
contact-line speed condition. This type of mode crossing possibly can be used to
explain instabilities observed experimentally (Vukasinovic et al., 2007).
The utility of the contact-line speed condition is that by varying the spreading
parameter Λ, one can smoothly ‘deform’ the problem between the two kinematic
disturbances, natural (Λ → 0) and pinned (Λ → ∞). Application of the contactline speed condition shows that at finite values of Λ, the oscillation frequencies
are damped, despite viscosity being zero. The lower-order mode shapes dissipate
the most energy over an oscillation cycle, unlike bulk viscous effects where the
higher-order modes dissipate more energy. This behavior can be understood using
the interpretation of the spreading parameter as a measure of the mobility of the
contact-line. According to this constitutive law, effective dissipation is related to
contact-line motion. Lower-order mode shapes have larger contact-line motions,
thus, one would expect their dissipation to be larger according to this constitutive law. Similarly, the sub-hemispherical drops have contact lines that are more
mobile, in comparison to the super-hemispherical ones and equivalently dissipate
more energy. In the super-hemispherical limit α → 180◦ (non-wetting), the contact
lines are essentially immobile and the distinction between the natural and pinned
disturbance is lost, resulting in frequencies that are indistinguishable.
In the following sections the mathematical problem associated with the small
123
!
"
Figure 5.2: Sessile drop equilibrium shape in polar view showing vectors
normal n and tangential t to the surface Γ.
deformations of the sessile drop is formulated. The disturbance classes are defined and the governing equations are reduced via a normal mode expansion to a
functional eigenvalue equation. The frequency spectrum is computed from a set of
algebraic equations, which results from the implementation of the Rayleigh-Ritz
procedure. The oscillation frequencies and/or growth rates depend upon the static
contact-angle α, azimuthal wave-number l and spreading parameter Λ. The dependence of the spectrum on these parameters is discussed and some concluding
remarks are offered.
5.2
Mathematical formulation
The sessile drop is a surface of constant mean curvature H and equivalently a
static equilibrium according to the Young-Laplace equation
p
= κ1 + κ2 ≡ 2H,
σ
124
(5.4)
(a)
(b)
(c)
z
G
ΗHs,j,tL
r
Α
x
Figure 5.3: Definition sketch with unperturbed Γ (dashed) and perturbed
interface η (solid) in (a) polar cross-section and three-dimensional
(b) perspective and (c) top views.
which relates the principal curvatures, κ1 and κ2 , to the pressure p there. This
equilibrium surface Γ may be defined parametrically as
1
sin(s) cos(ϕ),
sin(α)
1
Y (s, ϕ; α) = −
sin(s) sin(ϕ),
sin(α)
1
Z(s; α) =
(cos(s) − cos(α)) ,
sin(α)
X(s, ϕ; α) = −
(5.5a)
(5.5b)
(5.5c)
using arc-length s ∈ [−α, α] and azimuthal angle ϕ ∈ [0, 2π] as generalized surface
coordinates. As depicted in figure 5.2, the equilibrium surface is scaled by its base
radius r and parameterized by contact-angle α. Thus, the scaled drop volume is
given by
V /r3 =
³ π ´ 2 − 3 cos α + cos3 α
sin3 α
3
.
(5.6)
The interface is given a small perturbation η(s, ϕ, t) (c.f. figure 5.3). No domain
perturbation is needed for small deformations, thus the droplet domain
D ≡ {(x, y, z)| 0 ≤ x ≤ X(s, ϕ; α), 0 ≤ y ≤ Y (s, ϕ; α), 0 ≤ z ≤ Z(s; α)}
125
(5.7)
is bounded by a free surface ∂Df (≡ Γ) of uniform surface tension σ, and a planar
surface-of-support ∂Ds ;
∂Df ≡ {(x, y, z) | x = X(s, ϕ; α), y = Y (s, ϕ; α), z = Z(s; α)}, (5.8a)
∂Ds ≡ {(x, y, z) | z = 0}.
(5.8b)
The droplet is immersed in a passive gas and the effect of gravity is neglected.
5.2.1
Hydrodynamic field equations
The fluid is incompressible and the flow is assumed to be irrotational. Therefore,
the velocity field may be described as v = ∇Ψ, where the velocity potential Ψ
satisfies Laplace’s equation
∇2 Ψ = 0 [D]
on the drop domain.
(5.9)
Additionally, the velocity potential satisfies the no-
penetration condition
∇Ψ · ẑ = 0 [∂Ds ]
(5.10)
on the surface-of-support and a (linearized) kinematic condition
∂Ψ
∂η
=−
[∂Df ]
∂n
∂t
(5.11)
on the free surface, which relates the normal velocity to the perturbation amplitude
there. In the limit of small interface deflection and in accordance with potential
flow theory, the pressure field is expressed by the linearized Bernoulli equation
p=%
∂Ψ
[D],
∂t
(5.12)
where % is the fluid density. Finally, deviations from the equilibrium surface Γ
generate pressure gradients, and thereby flows, according to the Young-Laplace
126
equation
¡
¢
p/σ = −∆Γ η − κ21 + κ22 η [∂Df ].
(5.13)
Here the Laplace-Beltrami operator ∆Γ and principal curvatures κ1 , κ2 are defined
on the equilibrium surface and ∆Γ is given below by eqn. (5.17).
The hydrodynamic field equations (5.9)-(5.13) govern the motion of an inviscid
fluid, whose spherical-cap interface is given a small disturbance. Formally, the
field equations must be augmented with a boundary condition on the three-phase
contact-line to form a well-posed system of partial differential equations. Alternatively, one needs to specify the disturbance class to which the interface is subject.
5.3
Derivation of functional eigenvalue equation
To study the interfacial motion of the sessile drop, the hydrodynamic equations
are reduced via a normal mode expansion. The resulting eigenvalue problem is
then formulated as a functional equation on linear operators, from which the characteristic frequencies and corresponding mode-shapes are readily computed.
5.3.1
Normal-mode reduction
Normal modes
η(s, ϕ, t) = y(s)eiωt eilϕ , Ψ(x, t) = φ(ρ, θ)eiωt eilϕ ,
127
(5.14)
are applied to the field equations (5.9)-(5.13) to generate the following eigenvalue
problem,
µ ¶
µ
¶
1 ∂ ∂φ
1
∂
∂φ
l2
+
sin
θ
−
φ = 0 [D],
ρ2 ∂ρ ∂ρ
ρ2 sin θ ∂θ
∂θ
ρ2 sin2 θ
∂φ
= 0 [∂Ds ],
∂n
∂φ
= −iωy [∂Df ],
∂n
¶
µ 2
¢
∂φ ¡ 2
sin (α) ∂φ
2 ∂φ
2
−∆Γ
− κ1 + κ2
+l
= λ2 φ [∂Df ],
2
∂n
∂n
sin (s) ∂n
Z
∂φ
= 0,
Γ ∂n
%ω 2 r3
.
λ2 ≡
σ
(5.15a)
(5.15b)
(5.15c)
(5.15d)
(5.15e)
(5.15f)
The problem is parameterized by the azimuthal wave-number l. Equation (5.15a)
is Laplace’s equation written in spherical coordinates (ρ, θ), (5.15b) is the nopenetration condition on the support surface, (5.15c) is the kinematic condition
and (5.15e) is recognized as the integral form of the incompressibility condition or
a volume conservation constraint. The dynamic pressure balance across the free
surface is represented by (5.15d) with the principal curvatures given by
κ1 = κ2 = sin(α).
(5.16)
The Laplace-Beltrami operator
1 ∂
∆Γ y ≡ √
g ∂uα
µ
√
gg
αβ
∂y
∂uβ
¶
(5.17)
is defined by functions y on the equilibrium surface. Here the definition of the
surface metric


2
0
¡
¢2
 csc (α)

2
gαβ ≡ xα · xβ = 
 , g = sin(s)csc (α) ,
0
(csc(α) sin(s))2
allows one to write (5.15d) as
µ ¶00
µ ¶0 µ
¶µ ¶
∂φ
∂φ
l2
∂φ
λ2
+ cot(s)
+ 2−
φ,
=
−
∂n
∂n
∂n
sin2 (s)
sin2 (α)
128
(5.18)
(5.19)
where differentiation is with respect to the arc-length coordinate, 0 = d/ds.
5.3.2
Operator formalism
This integro-differential equation governs the motion of the interface and may be
formulated as a operator equation using one of two alternative representations.
Forward problem
The first formulation uses the interface deflection ∂φ/∂n as the unknown function,
giving rise to the following operator equation
¸
· ¸
∂φ
∂φ
λ2
2
; l = −λ̂ M
, λ̂2 ≡
,
K
∂n
∂n
sin2 (α)
·
(5.20)
which is referred to as the forward problem. Here M is an integral operator
representative of the fluid inertia and K a differential operator related to the
curvature,
·
¸
∂φ
M
≡ φ,
(5.21a)
∂n
·
¸ µ ¶00
µ ¶0 µ
¶µ ¶
∂φ
∂φ
∂φ
l2
∂φ
K
;l ≡
+ cot(s)
+ 2−
. (5.21b)
2
∂n
∂n
∂n
∂n
sin (s)
To proceed with this formulation, one must construct a sufficiently general solution
to the boundary value problem
∇2 ψ = 0 [D],
∂ψ
= fk [∂Df ].
∂n
(5.22)
Equivalently, given a surface deformation fk , one needs to compute the corresponding velocity potential ψk , in accordance with the inertia operator (5.21a).
In general, for all but the simplest of geometries, solution of this Neumann-type
129
boundary value problem requires a computationally intensive approach, such as a
boundary integral method.
Inverse problem
Alternatively, one may use the velocity potential φ as the unknown function and
define the inverse problem
M −1 [φ] = −λ̂2 K −1 [φ; l] ,
(5.23)
which follows directly from (5.19). Here the integro-differential nature of the governing equation persists, with
M −1 [φ] ≡
∂φ
∂n
(5.24)
the differential operator and K −1 an integral operator, inversely related to the
curvature operator K defined in (5.21b). As with the forward problem, the primary difficulties associated with the inverse problem are related to the integral
operator. Specifically, construction of the inverse operator K −1 depends on the
parametrization of the equilibrium surface and may or may not be analytically
tractable.
Forward vs. Inverse problem
The two representations defined here are completely equivalent, but each has their
respective difficulties. In the forward problem, one has to construct a sufficiently
general solution to Laplace’s equation with Neumann boundary conditions. In
most cases, this is analytically intractable. Likewise, one must construct the
130
Green’s function to the differential curvature operator to proceed with the inverse operator formalism. By specifying the relevant boundary conditions on the
three-phase contact-line, the disturbance class to which the interface is subject is
set and one may proceed with the most relevant operator formulation, either the
forward or inverse problem. While the inverse problem is seen to be more tractable
in general, the forward problem has an attractive variational structure that can be
exploited.
5.3.3
Contact-line conditions
To compute the spectrum of the eigenvalue problem (5.15), one needs to specify
the disturbance class, via boundary conditions on the three-phase contact-line.
Namely, ‘allowable’ solutions are decomposed into two distinct classes, kinematic
or dynamic, as mentioned above. Kinematic disturbances depend solely on the
interface deflection, whereas dynamic disturbances are related to ‘dynamic’ field
quantities, such as the contact-line speed. To ensure the boundary-value problem
is well-posed mathematically, the kinematic and dynamic boundary conditions are
augmented with the following restriction,
¯
∂φ ¯¯
− bounded,
∂n ¯s=0
a necessary conditions to guarantee the interface disturbance is physical.
131
(5.25)
(a)
(b)
z
z
G
G
Α
r
x
r
Α
x
Figure 5.4: Kinematic disturbances classes for the sessile drop: (a) natural
and (b) pinned contact-line.
Natural
The first type of kinematic disturbance is shown in figure 5.4(a) and preserves the
static contact-angle α in accordance with
∂
∂s
µ
∂φ
∂n
¶
¯
¯
∂φ
+ cos(α)
= 0¯¯ ,
∂n
s=α
(5.26)
and is termed the ‘natural’ boundary condition. A more thorough derivation of
the natural contact-line condition (A.9) is given in Appendix A.
Pinned
The second class of kinematic disturbance is shown in figure 5.4(b) and has ‘pinned’
contact-lines,
¯
∂φ ¯¯
= 0.
∂n ¯s=α
(5.27)
As noted by Courant & Hilbert (1953) for the general boundary-value problem, the
pinned contact-line is the most restrictive perturbation as the size of its function
space of allowable disturbances is the smallest.
132
Α
Observed
Continuous
uCL
Figure 5.5: Typical contact-line behavior observed (solid) and the continuous
constitutive law model (dashed) imposed here.
Dynamic contact-line
The dynamic contact-line condition follows by assuming that the contact-angle
depends smoothly on the contact-line speed through the function f (c.f. figure 5.5),
µ
µ ¶¶
∂φ
α + ²α̂ = f 0 + ² iω
.
(5.28)
∂n
This postulated constitutive law is inversely related to the Hocking contact-line
speed condition, which assumes that the contact-line speed is proportional to
the deviation from the static contact-angle (mentioned above). The variation in
contact-angle
µ
0
α̂ = iω f (0)
∂φ
∂n
¶
(5.29)
found from linearizing (5.28) is applied to (A.9), which results in the following
boundary condition
µ ¶
µ ¶
µ ¶¯
∂ ∂φ
∂φ
∂φ ¯¯
, Λ ≡ f 0 (0).
+ cos(α)
= iωΛ
∂s ∂n
∂n
∂n ¯s=α
(5.30)
Here 1/Λ is a measure of the mobility of the contact-line, which is used as a
parameter which can smoothly change the boundary condition from natural to
133
pinned. This is sometimes called a ‘homotopy’ parameter. In the limit Λ → 0,
(5.30) reduces to the natural condition (fully mobile) and Λ → ∞ corresponds
to the pinned contact-line condition (immobile). It should be noted that contactangle hysteresis is not allowable here by linearization of (5.28).
5.4
Solution method for kinematic disturbances
As stated earlier, the spectrum of the operator equation (5.19) may be computed
by either the forward or inverse method. This choice depends upon the imposed
disturbance class. Here the inverse method will be used for the kinematic disturbances, whereas it is advantageous to analyze the contact-line speed condition by
the forward method.
5.4.1
Inverse-operator construction
The two kinematic disturbances, either natural or pinned, are structurally similar and their respective inverse operators are derived simultaneously. To use the
inverse operator formalism, one must construct the integral operator
Z
K
−1
1
[φ] (x) =
G (x, y; l) φ (y) dy,
(5.31)
b
which is simply the Green’s function or fundamental solution of the curvature
operator (5.21b),

i
h


ξ(l)y1 (y; l) τ2 (l) y1 (x; l) − y2 (x; l) b < x < y < 1
τ1 (l)
G (x, y; l) =
h
i


ξ(l)y1 (x; l) τ2 (l) y1 (y; l) − y2 (y; l) b < y < x < 1.
τ1 (l)
134
(5.32)
Here the coordinate transformation x ≡ cos(s) has been used in the Green’s function, which is parameterized by azimuthal wave-number l and the transformed
contact-angle b ≡ cos(α). The functions y1 and y2 belong to the kernel of the
curvature operator K and are given by:
y1 (x; 0) = P1 (x), y2 (x; 0) = Q1 (x),
(1)
(1)
y1 (x; 1) = P1 (x), y2 (x; 1) = Q1 (x),
¶l/2
µ
1−x
,
y1 (x; l ≥ 2) = (x + l)
1+x
µ
¶l/2
(x + l)
1+x
y2 (x; l ≥ 2) =
,
2l (l2 − 1) 1 − x
(1)
(5.33a)
(5.33b)
(5.33c)
(1)
where P1 , Q1 and P1 , Q1 are the order 0 and 1 Legendre functions of index 1,
respectively. Similarly, the scale factor is given by



1/2 l = 1
ξ(l) ≡


1
l 6= 1,
(5.34)
while the parameters τ1 and τ2 are related to the contact-line boundary conditions
and expressed as
(n)
τ1
= y10 (b; l) + √
b
b
(n)
y1 (b; l), τ2 = y20 (b; l) + √
y2 (b; l),
2
1−b
1 − b2
(5.35)
for the natural and
(p)
(p)
τ1 = y1 (b; l), τ2 = y2 (b; l)
(5.36)
pinned contact-line disturbance classes, respectively.
Axisymmetric operator equation (l = 0)
Volume must be conserved by (5.15e) for this incompressible fluid. To satisfy this
condition, recall that the velocity potential φ is defined up to an arbitrary constant
135
C, which allows one to write (5.23) as
· Z 1
¸
Z 1
∂φ
2
(x) = −λ̂ C
G(x, y; 0)dy +
G(x, y; 0)φ(y)dy .
∂n
b
b
(5.37)
Integrating (5.37) along the equilibrium surface and enforcing (5.15e) uniquely
determines this constant
R 1R 1
C=−
G(x, y; 0)φ(y)dydx
b b
R 1R 1
b b
G(x, y; 0)dydx
.
(5.38)
Finally, the functional eigenvalue equation for the axisymmetric disturbance class
is written as
∂φ
(x) =
∂n
" R 1R 1
λ̂
2
G(x, y; 0)φ(y)dydx
b b
R 1R 1
G(x, y; 0)dydx
b b
Z
Z
1
G(x, y; 0)dy −
b
#
1
(5.39)
G(x, y; 0)φ(y)dy ,
b
with volume conserved for all φ.
Azimuthal-dependent operator equation (l 6= 0)
Unlike the axisymmetric operator equation (5.39), the volume conservation constraint (5.15e) is naturally satisfied for interface disturbances with azimuthal wavenumber l ≥ 1. Thus, the corresponding operator equation is given by
∂φ
(x) = −λ̂2
∂n
5.4.2
·Z
¸
1
G(x, y; l)φ(y)dy .
(5.40)
b
Rayleigh-Ritz
Stationary values of the operator equations (5.39,5.40) are the characteristic oscillation frequencies of the sessile drop, provided the no-penetration condition (5.15b)
is satisfied. This can be accomplished through proper selection of the potential
136
functions φ. Once these basis functions are selected, a Rayleigh-Ritz procedure is
used to reduce the operator equation into a standard algebraic eigenvalue equation.
The method is sketched here, while a more thorough illustration of the method is
given in Segel (1987). To begin, the eigenvalues λ of the operator equation
B [y] = λA [y]
(5.41)
are the stationary values of the functional
λ = min
(B[y], y)
, y ∈ S,
(A[y], y)
(5.42)
where S is a predetermined function space. A solution series,
y=
X
ci yi ,
(5.43)
i=1
constructed from functions yi ∈ S are applied to the functional (5.42) and minimized with respect to the coefficients ci to generate a set of algebraic equations
X
(bij − λaij ) cj = 0,
j=1
(5.44a)
Z
bij ≡
B[yi ] yj ,
(5.44b)
A[yi ] yj ,
(5.44c)
Z
aij ≡
from which the eigenvalues may be computed.
Axisymmetric operator (l = 0)
The necessary input for the Rayleigh-Ritz procedure is a solution series
φ=
N
X
aj φj ,
j=1
137
(5.45)
constructed from basis functions φj . These functions are applied to (5.39) and
inner products are taken to generate a set of linear algebraic equations
N ³
X
´
mij − λ̂2 κij aj = 0,
(5.46)
j=1
which have matrix elements
Z
1
mij ≡
b
µ
∂φi
∂n
¶
·
¸
G0i Gj0
(φj ) dx, κij ≡
− Gij ,
G00
(5.47)
with
·Z
1
Gij ≡
b
¶
¸
τ2
y1 (x; 0)φi (x)dx
y1 (x; 0) − y2 (x; 0) φj (x)dx
τ1
b
Z 1
Z 1
+
y1 (x; 0)φi (x)
y2 (y; 0)φj (y)dydx
b
x
Z 1
Z 1
−
y2 (x; 0)φi (x)
y1 (y; 0)φj (y)dydx.
¸ ·Z
1
µ
b
(5.48)
x
Allowable solutions of functional (5.39) necessarily satisfy the hydrodynamic
field equations (5.15). Recall, volume conservation (5.15e) has been satisfied by
proper selection of the constant C and the contact-line conditions are incorporated
into the Green’s function (5.32), but the no-penetration condition (5.15b) has yet
to be satisfied. This can be accomplished through proper selection of the basis
functions
φj (ρ, θ) = ρ2j P2j (cos θ) ,
(5.49)
given here in spherical coordinates, ρ and θ, and chosen to be harmonic, as required by (5.15a). Here P2j is the Legendre polynomial of degree 2j. The normal
derivatives of the basis functions (5.49), evaluated on the equilibrium surface, are
expressed as
∂φj
≡ ∇φj · n = 2jP2j (cos θ) (− sin s sin θ + cos s cos θ) (ρ)2j−1
∂n
0
+ sin θP2j
2j−1
(cos θ) (sin s cos θ + cos s sin θ) (ρ)
138
,
(5.50)
using mixed coordinates for efficiency in presentation. The basis functions (5.49)
and their normal derivatives (5.50) are defined on the equilibrium surface through
the following coordinate transformation,
ρ≡
√
X2
2
+Y +
Z 2,
Z
cos θ ≡ √
, sin θ ≡
X2 + Y 2 + Z2
r
X2 + Y 2
,
X2 + Y 2 + Z2
(5.51)
which relates spherical coordinates, whose origin is centered on the surface-ofsupport, to the arc-length coordinate. Here X = X(s), Y = Y (s), Z = Z(s) have
been defined in (5.5). Finally, the basis functions (5.49) are applied to (5.47) to
generate a set of algebraic equations (5.46), from which the characteristic oscillation frequencies are computed.
Azimuthal operator (l 6= 0)
An analogous solution series
φ(l) =
l+N
X
(l)
aj φj
(5.52)
j=l
is used to compute the stationary values of the functional (5.40), as they depend
(l)
upon the azimuthal wave-number l. As before, the basis functions φj are applied
to (5.40) and inner products are taken to generate the matrix equation
l+N ³
X
(l)
(l)
mij − λ̂2 κij
´
aj = 0
(5.53)
j=l
with
Z
(l)
mij
1
≡
b
Ã
(l)
∂φi
∂n
139
!
³
(l)
φj
´
dx
(5.54)
and
(l)
κij
≡
ξ(l)
·Z
1
b
¶
¸
τ2
(l)
y1 (x; l) − y2 (x; l) φj (x) dx
τ1
b
Z 1
Z 1
(l)
(l)
+
y1 (x; l)φi (x)
y2 (y; l)φj (y)dydx
Z b1
Z x1
(l)
(l)
−
y2 (x; l)φi (x)
y1 (y; l)φj (y)dydx.
¸ ·Z
(l)
y1 (x; l)φi (x) dx
1
µ
b
(5.55)
x
As stated earlier, volume conservation (5.15e) is naturally satisfied for interface
disturbances with a non-trivial (l 6= 0) azimuthal wave-number. Hence, the basis
functions
(l)
(l)
φj (ρ, θ) = ρj Pj (cos θ)
(5.56)
(l)
are used to generate the matrix elements (5.54,5.55). Here Pj
is the Legendre
function of order l and degree j. The following restriction on the polar j and
azimuthal l wave-umbers is needed to ensure the no-penetration conditions is satisfied: l + j = even. Additionally, a consistency condition requires j ≥ l. Thus,
the sum in (5.52) runs from l to l + N . These conditions are a result of well-known
(l)
properties of the Legendre functions Pj (MacRobert, 1967).
The normal derivatives of the basis functions (5.56) on the equilibrium surface
are given by
(l)
∂φj
(l)
≡ ∇φ(l) · n = 2jP2j (cos θ) (− sin s sin θ + cos s cos θ) (ρ)2j−1
∂n
0 (l)
2j−1
+ sin θP2j (cos θ) (sin s cos θ + cos s sin θ) (ρ)
(5.57)
.
As before, to define the basis functions (5.56) and their normal derivatives (5.57)
on the equilibrium surface, one uses the coordinate transformation (5.51) and the
parametric representation of the equilibrium surface (5.5).
140
5.4.3
Results for kinematic disturbances
The eigenvalues λk,l of (5.46,5.53), as they depend upon the contact-angle α and
azimuthal wave-number l, have been computed using a resolution of N = 10 basis
functions in the solution series (5.45,5.52) for both natural and pinned disturbance
classes. Computations show this particular truncation exhibits relative error of
0.1% for the first three eigenfrequencies. The eigenfunction or velocity potential
(k,l)
associated with the eigenfrequency λk,l and corresponding eigenvector aj
is given
by
φ(k,l) (x) =
l+N
X
(k,l) (l)
φj (x).
aj
(5.58)
j=l
Here eigenfrequencies/vectors are distinguished by polar k and azimuthal l wavenumbers. Similarly, the mode shape or interface deformation related to this potential function is expressed as
y
(k,l)
(x) =
l+N
X
Ã
(k,l)
aj
j=l
(l)
∂φj
∂n
!
(x).
(5.59)
Low azimuthal wave-number l = 0, 1
To distinguish between kinematic disturbance classes, figure 5.6 plots frequency
λk,l against contact-angle α for l = 0, 1 azimuthal wave-number. Here the natural
frequency is always smaller than the corresponding pinned case, which could have
been anticipated considering the fixed contact-angle disturbance is the ‘natural’
disturbance for the sessile drop. Sample mode shapes are given in figures 5.7,5.8
for pinned and natural disturbances, respectively. Recall that the mode shapes
are effectively normalized by the metric inherited from the denominator in the
Rayleigh-Ritz ratio–that is, per unit ’inertia.’
141
(a)
(b)
Λ1,1
Λ2,0
10
Ang
3
Ang
Pin
Pin
6
1
50
70
2
90
110
Α
130
50
70
90
(c)
110
Α
130
(d)
Λ3,1
Λ4,0
13
18
Ang
Ang
Pin
Pin
12
8
6
3
50
70
90
110
Α
130
50
70
90
(e)
110
Α
130
(f )
Λ5,1
Λ6,0
30
24
Ang
Ang
24
Pin
16
16
8
50
70
90
Pin
110
8
Α
130
50
70
90
110
Α
130
Figure 5.6: Frequency λk,l against contact-angle α for natural (Ang) and
pinned (Pin) modes [k, l] with low azimuthal wave-number l =
0, 1: (a) [1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f )
[6, 0]. Here Im[λ1,1 ] = 0 in all cases except α > 90◦ (c.f. figure 5.10). Note that the scalings of the frequency axis are different from (a) to (f ).
142
(a)
(b)
(c)
(d)
(e)
(f )
Figure 5.7: Pinned mode shapes [k, l] for hemispherical base-state α = 90◦ :
(a) [1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f ) [6, 0]
(polar view).
(a)
(b)
(c)
(d)
(e)
(f )
Figure 5.8: Natural mode shapes [k, l] for sub-hemispherical base-state α =
60◦ : (a) [1, 1] , (b) [2, 0] , (c) [3, 1] , (d) [4, 0] , (e) [5, 1] , and (f )
[6, 0] (polar view).
143
(a)
(b)
(c)
Figure 5.9: Contact-line mobility of the natural mode shape [k, l] = [5, 1] for
contact-angle (a) 60◦ , (b) 90◦ , and (c) 120◦ .
As the static contact-angle α → 180◦ or in the limit of the drop touching the
support plane along an infinitesimal generating circle, the frequencies for the two
kinematic disturbances are indistinguishable. In this limit, the natural disturbance
class degenerates into the pinned disturbance. To explain, one can compare the
characteristic motion around the contact-line for the natural disturbance class
to show that the super-hemispherical α > 90◦ drops has a relatively immobile
contact-line that is essentially fixed, while the contact-line of the sub-hemispherical
α < 90◦ drop is much more mobile. This feature is evident from figure 5.9, which
plots the k, l = 5, 1 natural mode shape for three different base-states. Here the
interface displacement around the contact-line is larger and more pronounced for
the sub-hemispherical drop in comparison to the super-hemispherical drop. In fact,
the displacement of the contact-line for the natural mode shape with polar wavenumber k decreases as the static contact-angle increases from 0◦ to 180◦ , where
the displacement is zero or the contact-line is fixed.
In addition to the static contact-angle, contact-line displacement is greatly
influenced by the polar wave-number k. Consider the displacement of the interface
at the contact-line for the natural mode shapes shown in figure 5.8, which is seen
to decrease as the polar wave-number increases. This contact-line behavior is
144
characteristic of the natural disturbance class and is independent of static contactangle. One could have also inferred that contact-line mobility decreases with polar
wave-number by examining the relative difference (λp −λn )/λp between the natural
(λn ) and pinned (λp ) frequencies, which may be used as a measure of contact-line
displacement (mobility). For example, the k, l = 6, 0 mode for the hemispherical
drop (α = 90◦ ) has a relative frequency difference of 11%, which compares to a
difference of 42% for the k, l = 2, 0 mode. A small difference in frequency (measure)
indicates the contact-line is essentially pinned and one cannot distinguish between
the natural and pinned disturbance. With regards to the example mentioned
above, one may conclude that the k, l = 6, 0 mode has a much smaller contact-line
displacement than the k, l = 2, 0 mode. Similarly, one could use this interpretation
for figure 5.6 to conclude that the mode shapes for the super-hemispherical basestates have relatively immobile contact-lines.
As might have been anticipated, the natural k, l = 1, 1 mode for the hemispherical base-state has zero-frequency and corresponds to a rigid, horizontal translation
of the drop’s center-of-mass. This motion is analogous to the zero-frequency mode
of the drop pinned on an equatorial circle-of-contact and likewise can be attributed
to the additional symmetry inherent in the hemispherical base-state (Field et al.,
1991). One can prove that the oscillation frequency of this mode shape is nec(1)
essarily zero. The proof is sketched here. To begin, the function P1 (x) is a
fundamental solution of the curvature operator (5.21b) and uniquely satisfies the
natural boundary conditions (5.26) for this base-state. Therefore, this function belongs to the kernel of (5.21b). It is also straightforward to show that the ‘inertia’
operator (5.21a) is positive definite. Thus, applying the Fredholm alternative for
linear operators to this particular mode shape delivers λ21,1 = 0.
145
-Λ21,1
0.0458
0.03
0.01
90
120 132.5
150
Α
180
Figure 5.10: Instability growth rate −λ21,1 against contact-angle α for a sessile
drop subject to a natural disturbance, exhibiting a maximum
growth rate (−λ21,1 = 0.0458) at α = 132.5◦ .
(a)
(b)
Figure 5.11: Typical instability mode shape [1, 1] with contact-angle α =
120◦ in (a) polar and (b) three-dimensional views.
‘Walking’ instability
In addition to a number of oscillatory mode shapes, the sessile drop exhibits instability (λ2 < 0) to the natural disturbance class for the super-hemispherical
base states; that is, those in the range of contact angles 90◦ < α < 180◦ . Figure 5.10, which plots the square of the instability growth rate −λ21,1 against static
146
(a)
(b)
(c)
(d)
(e)
(f )
Figure 5.12: Instability mode shape [1, 1] with contact-angle (a) 91◦ , (b) 105◦ ,
(c) 120◦ , (d) 135◦ , (e) 150◦ , and (f ) 170◦ in polar view.
(a)
(b)
(c)
(d)
(e)
(f )
Figure 5.13: Time series of the instability mode shape with largest growth
rate (−λ21,1 = 0.0458): (a, b, c) polar view and (d, e, f ) contactline footprint at time (a, d) t = 0, (b, e) t = T /2 and (c, f ) t = T
(α = 132.5◦ ). Here T represents the time it takes to disturb the
interface by a given amplitude (T = 1.226 in this figure).
147
contact-angle α, shows that the maximum growth rate (−λ21,1 = 0.0458) occurs
at α = 132.5◦ . Typical unstable mode shapes are shown in figures 5.11 & 5.12.
Here the disturbed shape (solid) has been made to intersect the undisturbed shape
(dashed) at its apex (north pole) as a convention, although there is no fundamental
reason to do so (c.f. figure 5.12). The characteristic feature of this instability is an
advancing contact-line, which is accelerating relative to its corresponding receding
contact-line, thereby displacing mass from one side of the drop to the other, as
shown in figure 5.13. To illustrate, figure 5.13(b) demonstrates how the contactline footprint deforms from a circle into an ellipse as the instability evolves in time.
Forcing the contact-lines to be ‘pinned’ naturally suppresses the instability and the
possibility of ‘walking’, while generating oscillatory motion for all contact angles
(c.f. figure 5.6(a)).
Instability occurs when the perturbed state has a lower potential energy than
the corresponding base-state. To investigate the nature of this instability, one can
start with the second variation of potential energy (c.f. eqn. 1.5),
µ
¶¸
Z 1·
¡¡
¢ 0 ¢0
1
2
2
δ U =−
1−x y + 2−
y dx,
1 − x2
b
(5.60)
and integrate by parts using the natural boundary condition (5.26) to obtain
¶ ¸
µ
Z 1·
√
¡
¢ 0 2
1
2
2
2
1 − b2 y 2 (b),
(5.61)
δ U=
y
dx
+
b
1 − x (y ) − 2 −
2
1−x
b
with b ≡ cos α. This equivalent form of the total disturbance energy Et may be
decomposed into two interfacial energies (E1 , E2 ) and one contact-line energy (E3 ),
Et = E1 + E2 + E3
µ
¶ ¸
Z 1·
¡
¢ 0 2
1
2
1 − x (y ) +
y 2 dx
E1 ≡
2
1−x
b
Z 1
E2 ≡ −
2 y 2 dx
√b
E3 ≡ b 1 − b2 y 2 (b).
148
(5.62a)
(5.62b)
(5.62c)
(5.62d)
E
Et
E1
E2
E3
4
3
2
1
-1
-2
-3
-4
30
60
90
120
150
180
Α
Figure 5.14: Decomposition of disturbance energy (E) against contact-angle
α for the [1, 1] instability mode.
Here the interfacial energy E1 is a positive-definite (stabilizing) measure of the
gradients in mean-curvature of the perturbed surface, while E2 is negative-definite
(destabilizing) and represents the tendency of a volume of liquid to form an isolated
spherical drop (minimal energy state). In contrast, the contact-line energy E3 can
either stabilize (E3 > 0) or destabilize (E3 < 0), depending upon the geometry of
the base-state. Contrast with the case of the rivulet reported by Davis (1980). See
also, Chapter 6.
Of course, a linear analysis cannot reveal what configuration that an instability
will finally lead to, but one may speculate that the instability reported here will
cause the drop to walk along the surface-of-support because of the following mechanism. Figure 5.14, which plots the decomposition of disturbance energy (5.62) for
the instability mode shape y, shows that both interfacial (E2 ) and contact-line (E3 )
terms drive the instability. The interfacial energy E2 is always destabilizing and
its associated instability mechanism has been mentioned above. In contrast, the
149
(a)
(b)
Figure 5.15: Visualization of the left (Fl ) and right (Fr ) constraint force at
the contact-line for the natural mode shapes with (a) even and
(b) odd symmetry about the vertical mid-plane.
contact-line energy E3 is destabilizing for the range of contact angles that exhibit
instability (90◦ < α < 180◦ ). Here energy is supplied to the drop from the contactline by a ‘virtual’ force acting at the contact-line that does work on the drop. To
explain the origins of this force, consider the natural boundary condition, which
preserves the static contact-angle, as an auxiliary constraint on the drop motion.
An arbitrary (unconstrained) interface disturbance does not satisfy the static force
balance at the contact-line or preserve the static contact-angle and is therefore not
in equilibrium. Thus, to maintain the static contact-angle during deformation,
a constraint force must be applied on the contact-line. This constraint force is
typical of problems with an auxiliary condition and may or may not have a net
effect on the drop. For example, if the disturbance is mirror-symmetric (even)
about the vertical mid-plane (c.f. figure 5.15(a)), the constraint forces from the
left (Fl ) and right (Fr ) contact lines produce no net force on the drop, because
they have equal magnitude and act in opposite directions. On the contrary, disturbances with odd-symmetry (c.f. figure 5.15(b)) can generate a net force, because
150
the constraint forces from the advancing and receding contact-line act in the same
direction. This net force drives the ‘walking’ instability and also explains why the
instability mode shape is the sole carrier of the drop’s horizontal center-of-mass
motion.
High azimuthal wave-number l ≥ 2
Lastly, one can examine the role of azimuthal wave-number l on the characteristic frequencies and mode shapes for the kinematic disturbance classes. The mode
shapes are decomposed into three categories and are conveniently distinguished by
their wave-number pair [k, l] using standard terminology for the spherical harmonics (MacRobert, 1967); i) zonal, ii) sectoral, and iii) tesseral. Zonal shapes (l = 0)
are axisymmetric and do not intersect the undisturbed shape along a longitudinal
line. On the contrary, sectoral mode shapes (k = l) have no latitudinal crossings, except if the contact lines are pinned. Mode shapes associated with all other
wave-number pairs are referred to as tesseral. Given a natural mode shape with
wave-number pair [k, l], there are precisely 2l longitudinal and k − l latitudinal
crossings or intersections of the undisturbed shape. For the sectoral and tesseral
mode shapes, there is necessarily a crossing at the north pole that has not been
counted. The pinned mode shapes have two additional latitudinal crossings, because the support plane lies along a constant latitude. To illustrate, mode shapes
from the three categories are plotted in figures 5.16 & 5.17 for natural and pinned
contact-line disturbances, respectively.
With regards to oscillation frequencies, figure 5.18 shows that the natural frequencies of polar wave-number k are degenerate with respect to azimuthal wavenumber l for the hemispherical base-state α = 90◦ . Varying the static contact-angle
151
(a)
(b)
(c)
Figure 5.16: Natural mode shapes [k, l] in polar and three-dimensional
side/top views for (a) zonal [8, 0] , (b) sectoral [5, 5] , and (c)
tesseral [7, 3] disturbances for α = 90◦ .
α or volume breaks this degeneracy. Here frequencies for the sub-hemispherical
α < 90◦ base-states decrease with increasing azimuthal wave-number. The opposite is true for the super-hemispherical α > 90◦ base-states. As shown, the relative
frequency difference between the zonal l = 0 and sectoral l = k mode shapes
can be large. For example, the frequency difference between the k, l = 4, 0 and
k, l = 4, 4 mode shapes approaches 100% for the base-state α = 50◦ subject to
natural disturbances. Although the magnitude may be surprising, the difference
in frequencies is not completely unexpected. Recall that the sectoral mode shape
has the fewest number of latitudinal crossings or ‘nodes’, has the greatest contactline mobility and is therefore the preferred disturbance by sub-hemispherical basestates. In contrast, super-hemispherical base-states favor less mobile contact lines
152
(a)
(b)
(c)
Figure 5.17: Pinned mode shapes [k, l] of the hemispherical base-state α =
90◦ in polar and three-dimensional side/top views for (a) zonal
[4, 0] , (b) sectoral [3, 3] , and (c) tesseral [5, 1] disturbances.
and zonal mode shapes are to be expected. The qualitative behavior of the two
types of kinematic disturbance are similar, but there are a couple of distinctions.
The first and most apparent difference is the magnitude of the pinned contact-line
frequency splitting between the zonal and sectoral mode shapes, which is much
smaller than the corresponding frequency range for the natural disturbance class.
The size of this splitting can be seen in figure 5.19, which plots the pinned frequencies against static contact-angle. Another interesting feature of figure 5.19 is
that the azimuthal wave-number degeneracy of the hemispherical base-state has
been eliminated. The relationship between this degeneracy and the symmetry of
the system are intimately connected and will be expanded upon further in the
following section.
153
(a)
(b)
Λ4,l
Λ5,l
l‡0
l‡2
l‡4
14
l‡1
l‡3
l‡5
22
18
10
14
10
6
6
50
70
90
110
Α
130
50
70
90
(c)
Λ6,l
Λ7,l
26
18
12
10
6
70
90
110
Α
130
50
70
90
(e)
Λ8,l
40
30
l‡0
l‡2
l‡4
l‡6
l‡8
90
Α
130
Λ9,l
45
35
l‡1
l‡3
l‡5
l‡7
l‡9
25
10
70
110
(f )
20
50
l‡1
l‡3
l‡5
l‡7
34
18
50
Α
130
(d)
l‡0
l‡2
l‡4
l‡6
24
110
110
15
Α
130
50
70
90
110
Α
130
Figure 5.18: Natural frequency λk,l , as it depends upon azimuthal wavenumber l, against contact-angle α for fixed polar wave-number
(a) k = 4, (b) k = 5, (c) k = 6, (d) k = 7, (e) k = 8, and (f )
k = 9.
154
(a)
(b)
Λ4,l
Λ5,l
l‡0
l‡2
l‡4
18
14
l‡1
l‡3
l‡5
24
20
16
10
12
6
50
70
8
90
110
Α
130
50
70
90
(c)
Λ7,l
l‡0
l‡2
l‡4
l‡6
32
24
28
20
8
90
110
12
Α
130
50
70
90
(e)
Λ8,l
45
35
Λ9,l
l‡0
l‡2
l‡4
l‡6
l‡8
55
45
35
90
Α
130
l‡1
l‡3
l‡5
l‡7
l‡9
25
15
70
110
(f )
25
50
l‡1
l‡3
l‡5
l‡7
36
16
70
Α
130
(d)
Λ6,l
50
110
15
110
Α
130
50
70
90
110
Α
130
Figure 5.19: Pinned frequency λk,l , as it depends upon azimuthal wavenumber l, against contact-angle α for fixed polar wave-number
(a) k = 4, (b) k = 5, (c) k = 6, (d) k = 7, (e) k = 8, and (f )
k = 9.
155
5.5
Solution method for contact-line speed relation
In principle, the inverse method could be used to satisfy the dynamic contact-line
condition (5.30), but this formulation gives rise to an integral operator with the
eigenvalue parameter in its kernel, which is notorious for computational instability
(e.g. Walter, 1973). To circumvent this issue, select results from the natural disturbance class are utilized and the structure of the variational problem associated
with (5.19) is exploited. Essentially this is an ad hoc approach. It will be shown
that contact-angle variation associated with the contact-line speed condition (5.30)
is a purely dissipative process.
5.5.1
Operator construction
By exploiting the structure of the variational problem associated with (5.19), the
reduced operator equation is cast in a particularly advantageous form. Namely,
the second variation of surface energy is given by
¶
Z µ
¡ 2
¢
∂φ ∂φ
2
2 ∂φ
δ U =−
+ ∆Γ
κ1 + κ2
∂n
∂n ∂n
Γ
¶
Z µ µ ¶
∂ ∂φ
∂φ ∂φ
+
+ cos(α)
,
∂s ∂n
∂n ∂n
γ
(5.63)
which is comprised of free surface Γ and contact line γ contributions. The EulerLagrange equations associated with this second variation give rise to the hydrodynamic boundary value problem (5.15d) for the natural disturbance class. It is
clear from the contact-line contribution to (5.63) that the fixed contact-angle condition (5.26) is the ‘natural’ boundary condition, as there is no contact-line energy
associated with these motions. Thus, the natural disturbances are seen to be the
true minimizers of the functional eigenvalue problem (5.20) and the function space
156
spanned by these orthogonal mode shapes form a complete set.
5.5.2
Hybrid Ritz method
Recall that the necessary input to the Rayleigh-Ritz procedure is a set of functions
which span a pre-determined function space. Thus, in implementing the forward
method on the operator equation (5.20) one could choose to use the complete set of
natural mode shapes y (j,l) , defined in (5.59), as the basis functions in the requisite
solution series
l+N
∂φ(l) X (j,l)
=
cj y (x).
∂n
j=l
(5.64)
By construction, volume is conserved according to (5.15e) for this particular set
of functions. Application of the forward method requires the computation of the
velocity potential corresponding to the respective interface perturbation. As stated
earlier, this proves a daunting task for most geometries, including the spherical
cap geometry used here. However, judicious choice of the solution series (5.64)
can completely eliminate this difficulty. To explain, recall that application of the
inverse method to the operator equation (5.23) results in eigenfunctions (5.58) that
are identified as velocity potentials, with the corresponding interface disturbance
(5.59). That is, the eigenfunction (5.58) is a velocity potential φ that generates
an interface disturbance y ∝ ∂φ/∂n as defined in (5.59). Equivalently, one could
say the known interface disturbance y (j,l) generates the known velocity potential
φ(j,l) . This solves precisely the difficulty mentioned above: compute the velocity
potential for a given interface disturbance in a relatively simple manner. Of course
the question remains, why choose the natural mode shapes as the basis functions
for the solution series (5.64)? As stated earlier, the fixed contact-angle disturbances
are the ‘natural’ disturbances and the space which they span is the largest and
157
least restrictive (see Courant & Hilbert, 1953; Lanczos, 1986; Segel, 1987).
Finally, the set of algebraic equations which result from the application of the
Rayleigh-Ritz procedure to the forward problem are written as
l+N ·
X
(l)
Kij
j=l
¸
iω
ω2
(l)
(l)
+√
Φij +
M
cj = 0,
1 − b2 ij
1 − b2
(5.65)
with matrix elements
Z
(l)
Kij
1
≡
µ
¡
b
1−x
2
¢
µ
(i,l)
yxx
−
l2
+ 2−
1 − x2
2xyx(i,l)
(l)
Φij
≡ Λ y (i,l) (b) y (j,l) (b) ,
Z 1
(l)
φ(i,l) y (j,l) dx.
Mij ≡
¶
¶
y
(i,l)
y (j,l) dx, (5.66a)
(5.66b)
(5.66c)
b
As evident from the damped harmonic oscillator structure of (5.65), contact-angle
variation controlled by the spreading parameter Λ in (5.66b) is a purely dissipative
process.
5.5.3
Results
The oscillation frequencies are computed as the roots of the characteristic equation
resulting from (5.65) for a given truncation N . Given a static contact-angle α and
spreading parameter Λ, computations show that a truncation of N = 12 terms in
the solution series (5.64) is sufficient to generate relative error of 0.1% for the first
three frequencies. The mode shapes corresponding to these frequencies are then
given by
ψ
(k,l)
=
l+N
X
(k,l) (j,l)
cj
y
,
(5.67)
j=l
(k,l)
where cj
is the respective coefficient eigenvector with polar k and azimuthal l
wave-numbers.
158
(a)
(b)
Ωk,0
25
(c)
Qk,0
Γk,0
5
k‡2
k‡4
k‡6
20
15
k‡2
k‡4
k‡6
4
3
10
2
5
1
0.01
0.1
1
10
100
L
0.01
0.1
(d)
10
100
L
k‡2
k‡4
k‡6
15
5
10
100
L
0.01
1.0
0.5
0.1
1
10
100
L
k‡2
k‡4
k‡6
1
10
100
1.5
0.01
1
10
100
L
k‡2
k‡4
k‡6
2.0
1.5
1.0
0.5
0.5
0.1
0.1
(i)
1.0
0.01
0.01
Qk,0
2.0
L
L
1.5
Γk,0
2.5
k‡2
k‡4
k‡6
100
k‡2
k‡4
k‡6
2.0
(h)
14
12
10
8
6
4
2
10
2.5
k‡2
k‡4
k‡6
(g)
Ωk,0
1
(f )
3.5
3.0
2.5
2.0
1.5
1.0
0.5
10
1
0.1
Qk,0
Γk,0
0.1
0.01
(e)
Ωk,0
20
0.01
1
k‡2
k‡4
k‡6
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.1
1
10
100
L
0.01
0.1
1
10
100
L
Figure 5.20: Oscillation frequency ωk,0 (a, d, g), decay rate γk,0 (b, e, h) and
effective dissipation Qk,0 (c, f, i) as a function of the spreading
parameter Λ for a contact-angle (a, b, c) α = 75◦ , (d, e, f ) α =
90◦ and (g, h, i) α = 105◦ .
Effective dissipation
The spreading parameter Λ is a measure of the mobility of the contact-line and
can also be used to smoothly vary the boundary conditions on the three-phase
contact-line, thereby, changing the disturbance class from natural (Λ → 0) to
pinned contact-line (Λ → ∞). The postulated constitutive law (5.28), which
assumed the dynamic contact-angle was a function of contact-line speed gave rise
to the contact-line speed boundary condition (5.30) and finally the variational form
of the governing equations (5.65), from which it has been shown that contact-angle
159
variation is a purely dissipative process controlled by the spreading parameter. The
complex frequencies λ = −γ + iω are computed from (5.65) as a function of the
spreading parameter Λ for the axisymmetric (l = 0) mode shapes in figure 5.20.
Here in both the natural (Λ → 0) and pinned (Λ → ∞) limits, no energy is
dissipated for such motions, while for finite values of the spreading parameter, it
is the high polar wave-number mode shapes that have the largest decay rate γk,0 .
A similar decay rate relationship, γ ∝ (k − 1)(2k + 1), has been derived from bulk
viscous effects for the isolated spherical drop by Lamb (1932). That is, the decay
rate increases monotonically with polar wave-number k. However, if one uses the
damped-harmonic oscillator structure of (5.65) and related formalism to define the
effective dissipation over an oscillation cycle as
Qk,l ≡ 2π
γk,l
,
ωk,l
(5.68)
the results may be surprising. For reference, the dissipation due to bulk viscous
√
effects for the spherical fluid drop is given by Qk,l ∝ k.
As shown in figure 5.20(c, f, i), which plots the effective dissipation Qk,0 for
the axisymmetric mode shapes (l = 0) as a function of the spreading parameter
Λ, it is the low polar wave-number modes that dissipate the most energy. This
can be directly attributed to the contact-line mobility, which was discussed in a
prior section. Recall that the motion of the contact-line was most pronounced for
the low polar wave-number mode shapes. Thus, if the dissipation mechanism for
this constitutive law is the contact-line mobility, it stands to reason that the low
wave-number mode shapes will dissipate the most energy. Computational results
substantiate this interpretation. Furthermore, it has been established in a previous
section that contact-line mobility of the natural disturbance class decreases as the
static contact-angle increases. To wit, figures 5.20(c) and 5.20(i) plot the effective
dissipation Qk,0 vs. spreading parameter Λ for a sub-hemispherical (α = 75◦ ) and
160
(a)
(b)
Ωk,3
Γk,3
7
k‡3
k‡5
k‡7
25
20
(c)
Qk,3
4
k‡3
k‡5
k‡7
6
5
4
15
2
3
10
2
5
1
0.01
0.05 0.10
0.50 1.00
5.00
L
k‡3
k‡5
k‡7
3
1
0.05 0.10
0.50 1.00
5.00
L
0.05 0.10
0.50 1.00
5.00
L
Figure 5.21: Oscillation frequency ωk,3 (a), decay rate γk,3 (b) and effective
dissipation Qk,3 (c) as a function of the spreading parameter Λ
for contact-angle α = 75◦ and azimuthal wave-number l = 3.
(a)
Ω7,l
30
28
26
(b)
Γ7,l
7
l‡3
l‡5
l‡7
(c)
Q7,l
2.0
l‡3
l‡5
l‡7
6
5
4
24
1.0
3
22
2
20
1
0.05 0.10
0.50 1.00
5.00
L
l‡3
l‡5
l‡7
1.5
0.5
0.05 0.10
0.50 1.00
5.00
L
0.01
0.05 0.10
0.50 1.00
5.00
L
Figure 5.22: Oscillation frequency ω7,l (a), decay rate γ7,l (b) and effective
dissipation Q7,l (c) as a function of the spreading parameter Λ
for contact-angle α = 75◦ and polar wave-number k = 7.
super-hemispherical (α = 105◦ ) base-state, respectively. Unsurprisingly, the mode
shapes with smaller contact-angle α dissipate the most energy, as they have the
more mobile contact lines, consistent with the interpretation of this constitutive
law. For these comparisons, recall that the kinetic energy of the disturbance is
normalized via the denominator in Rayleigh-Ritz quotient.
Large relative dissipation of energy by the low polar wave-number modes is not
unique to the axisymmetric l = 0 disturbance class and persists for mode shapes
of all non-zero azimuthal wave-number. This feature of the contact-line speed
condition is independent of static contact-angle and is corroborated by figure 5.21,
which plots the complex frequencies and effective dissipation as a function of the
161
(a)
(b)
Ω6,l
17
16
Γ6,l
4
l‡2
l‡4
l‡6
3
15
(c)
Q6,l
l‡2
l‡4
l‡6
1.5
1.0
l‡2
l‡4
l‡6
2
14
0.5
1
13
0.05 0.10
0.50 1.00
5.00
L
0.05 0.10
0.50 1.00
5.00
L
0.05 0.10
0.50 1.00
5.00
L
Figure 5.23: Oscillation frequency ω6,l (a), decay rate γ6,l (b) and effective
dissipation Q6,l (c) as a function of the spreading parameter Λ
for contact-angle α = 105◦ and polar wave-number k = 6.
spreading parameter for a drop with sub-hemispherical base state α = 75◦ and fixed
azimuthal wave-number l = 3. Alternatively, one can fix the polar wave-number k
and analyze the effect of azimuthal wave-number l. To that end, figures 5.22 & 5.23
plot the complex frequency and effective dissipation against spreading parameter
Λ for the sub-hemispherical α = 75◦ base-state with l = 7 and super-hemispherical
α = 105◦ base-state with l = 6, respectively. Here no universal statement can be
made with regards to the dissipation of an arbitrary mode shape with fixed polar
wave-number. At different values of the spreading parameter, the mode shape that
dissipates the largest amount of energy may change. For example, figure 5.23(c)
shows that for spreading parameters Λ = 0.5 and Λ = 1, the [6, 4] and [6, 2] modes
have the largest effective dissipation, respectively. However, the maximal effective
dissipation over the range of spreading parameters is sensitive to the static contactangle and equivalently the volume of the base-state. For fixed polar wave-number
k, the mode shape of maximal dissipation has either the largest or smallest possible
azimuthal wave-number for the sub-hemispherical and super-hemispherical basestates, respectively.
162
Lc
100
Stable
1
0.01
Unstable
10-4
10-6
90
120
150
Α
180
Figure 5.24: Stability diagram for the [k, l] = [1, 1] mode: critical spreading
parameter Λc against contact-angle α.
‘Walking’ instability
The effective dissipation, which results from the dynamic contact-line condition,
can suppress the instability that the [1, 1] mode exhibits for the super-hemispherical
base-states (c.f. figure 5.10). To explain, recall that instability results when the
perturbed state has a lower energy than the base-state. The base-state may be
stabilized if there exists a mechanism to counteract this energy difference, such
as the effective dissipation from the dynamic contact-line condition, which is controlled by the spreading parameter Λ. Figure 5.24 plots the critical spreading
parameter Λc , where the base-state is neutrally stable or the dissipation exactly
balances the energy difference, against the static contact-angle α. In the singular
limit α → 180◦ , where the scaling radius shrinks to zero and the natural disturbance degenerates into a pinned contact-line, the instability window remains open
(c.f. figure 5.24) despite the small instability growth rate observed in this limit
(c.f. figure 5.10). Here the stabilizing (E > 0) effective dissipation (5.66b) tends
to zero and cannot overcome the destabilizing disturbance energy (Et < 0) seen in
163
(a)
(b)
Γ1,1
Γ1,1
1.5
0.02
1
0.00
-0.02
0.5
0.005
0.010
0.015
L
0.020
-0.04
-0.06
0.05 0.1
0.5 1
5
10
L
-0.08
Figure 5.25: Complex frequency λ = −γ + iω for the [1, 1] mode: (a) decay
rate γ1,1 > 0 and (b) instability growth rate γ1,1 < 0 as a function of the spreading parameter Λ for contact-angle α = 105◦ .
See figure 5.27 (a) for corresponding oscillation frequency ω1,1 .
figure 5.14. With regards to growth rate, the effective dissipation is a monotonically increasing function of the spreading parameter Λ, therefore the maximum
instability growth rate corresponds to Λ = 0. As such, the maximum growth rate
as a function of the contact-angle is shown in figure 5.10. A typical transition
from instability γ < 0 to damped oscillations γ > 0 for the [1, 1] mode is shown in
figure 5.25.
Symmetry breaking
As stated earlier, the natural mode shapes are degenerate with respect to azimuthal
wave-number for the hemispherical drop. This occurs for the hemispherical basestate because it possesses more configurational symmetry than the other basestates and symmetries are known to produce degeneracies via Noether’s theorem.
The variational formulation of this problem is consistent with Noether’s theorem
and degeneracies in the hemispherical limit are to be expected. As shown earlier in
164
figures 5.18 & 5.19, this degeneracy is lifted by either i) pinning the contact lines or
ii) by ‘breaking’ the configurational symmetry of the base-state. The degeneracy
breaking mechanism for ii) is straightforward, while an interpretation of i) is more
subtle. To elucidate, one can examine the second variation of surface energy (5.63)
and more specifically the ‘disturbance’ energy associated with the contact-line
motion (i.e. the second integral in (5.63)). Interface perturbations that have
no related contact-line disturbance ‘energy’ are deemed ‘natural’ and necessarily
satisfy the ‘natural’ boundary conditions for the system. For the sessile drop, the
natural disturbance preserves the static contact-angle of the sessile drop. Implicit
in Noether’s theorem is the assumption that the boundary conditions associated
with the second variation of energy are the ‘natural’ ones. Therefore, to lift the
degeneracy associated with the hemispherical base-state one can introduce small
deviations to the natural boundary conditions.
The utility of the contact-line speed condition (5.30) is that one can lift the degeneracy of the hemispherical drop by smoothly varying the spreading parameter
Λ. Figure 5.26 plots the computed oscillatory frequency ωk,l vs. spreading parameter Λ for the hemispherical base-state (α = 90◦ ). As expected, the degeneracy
persists in the natural limit Λ = 0, is broken at some finite value of spreading parameter and stays broken until the pinned contact-line limit is reached at Λ → ∞.
For fixed polar wave-number k, the mode shapes with largest azimuthal wavenumber l = k have the greatest frequency in the pinned contact-line limit, because
they have the smallest number of polar nodes, which makes it difficult to satisfy
the pinned contact-line condition. To illustrate this phenomenon, sample mode
shapes are shown in figures 5.16 & 5.17. Also, one can quantify the degree to
which the degeneracy is lifted by analyzing the relative difference in frequencies
between mode shapes of equal polar wave-number but differing azimuthal wave-
165
(a)
(b)
Ω1,l
2.5
2.0
1.5
Ω2,l
5.0
4.5
l‡1
4.0
1.0
3.5
0.5
3.0
0.1
0.2
0.5
1.0
2.0
5.0
L
l‡0
l‡2
0.1
0.2
0.5
(c)
7.5
2.0
5.0
2.0
5.0
2.0
5.0
L
(d)
Ω3,l
8.5
8.0
1.0
Ω4,l
12
l‡1
l‡3
11
7.0
l‡0
l‡2
l‡4
10
6.5
6.0
9
5.5
0.1
0.2
0.5
1.0
2.0
5.0
L
0.1
0.2
0.5
(e)
14
L
(f )
Ω5,l
15
1.0
Ω6,l
l‡1
l‡3
l‡5
19
18
l‡0
l‡2
l‡4
l‡6
17
13
16
12
0.1
0.2
0.5
1.0
2.0
5.0
15
L
0.1
0.2
0.5
1.0
L
Figure 5.26: Oscillation frequency ωk,l , as it depends upon azimuthal wavenumber l, against spreading parameter Λ for the hemispherical
(α = 90◦ ) base-state with polar wave-number (a) k = 1, (b)
k = 2, (c) k = 3, (d) k = 4, (e) k = 5, and (f ) k = 6.
166
(a)
(b)
Ω1,l
Ω2,l
4.5
2.0
1.5
4.0
l‡1
3.5
1.0
3.0
0.5
0.1
l‡0
l‡2
2.5
0.2
0.5
1.0
2.0
5.0
L
0.1
0.2
0.5
(c)
6.0
2.0
5.0
2.0
5.0
2.0
5.0
L
(d)
Ω3,l
7.0
6.5
1.0
Ω4,l
10
l‡1
l‡3
9
5.5
l‡0
l‡2
l‡4
8
5.0
7
4.5
0.1
0.2
0.5
1.0
2.0
5.0
L
0.1
(e)
Ω5,l
Ω6,l
17
l‡1
l‡3
l‡5
13
12
16
0.2
0.5
1.0
L
(f )
l‡0
l‡2
l‡4
l‡6
15
11
14
10
13
0.1
0.2
0.5
1.0
2.0
5.0
L
0.1
0.2
0.5
1.0
L
Figure 5.27: Oscillation frequency ωk,l , as it depends upon azimuthal wavenumber l, against spreading parameter Λ for a typical superhemispherical (α = 105◦ ) base-state with polar wave-number
(a) k = 1, (b) k = 2, (c) k = 3, (d) k = 4, (e) k = 5, and (f )
k = 6.
number. As shown in figure 5.26, the maximum relative frequency difference is
approximately 10% for the sectoral mode shapes (i.e. modes with k = l).
The degeneracy of the hemispherical sessile drop can be lifted by breaking
167
the configurational symmetry of the base-state, as can be seen in figures 5.18
& 5.19. One can study the effect of simultaneously applying the two degeneracy lifting mechanisms by considering the contact-line speed condition to either a
sub-hemispherical or super-hemispherical base-state. These oscillation frequencies
have been plotted against the spreading parameter Λ in figure 5.27 for the superhemispherical base-state, α = 105◦ . One can directly compare figures 5.26 & 5.27
to show that deviations in volume away from the hemispherical base-state, controlled by the static contact-angle, lead to a larger magnitude in frequency shift
amongst mode shapes of identical polar wave-number. Also, the degree of frequency splitting is much larger and more pronounced for the natural limit Λ → 0
than it is for the pinned contact-line limit Λ → ∞. For example, the frequency
difference between the k, l = 6, 0 and k, l = 6, 6 modes approaches 25% for the natural disturbance, while this difference is only 14% when the contact-line is fixed.
In addition, the transition from natural to pinned disturbance occurs at different
values of the spreading parameter. Moreover, for fixed polar wave-number k there
is the possibility that mode shapes of varying azimuthal wave-number l will oscillate with either a free-like or pinned-like contact-line at some particular value of
the spreading parameter. Consider figure 5.27(f ), the [6, 0] and [6, 6] modes are in
the natural and pinned regime at Λ = 0.8, respectively.
Mode crossings
One can imagine that as the static contact-angle is increased there is the possibility
of mode crossings, whereby two mode shapes might share the same characteristic
frequency (i.e. the classic ordering of the mode shapes could be distorted). Classically, the number of ‘nodes’ or zeroes of a given eigenfunction and the spectral
ordering of the corresponding eigenvalue are coincident. For example, the eigen168
Ωk,l
22
k,l=8,0
k,l=7,7
k,l=7,5
k,l=7,3
k,l=7,1
21
20
19
18
17
16
0.1
0.2
0.5
1.0
2.0
5.0
L
10.0
Figure 5.28: Modal crossings from the dynamic contact-line condition: oscillation frequency ωk,l as measured by the spreading parameter Λ
for α = 105◦ .
function which corresponds to the third numerical eigenvalue has three nodes. To
examine the possibility of these mode crossings, one can independently vary either
one or both of the degeneracy lifting mechanisms; the static contact-angle α or the
spreading parameter Λ. The latter is most applicable to systems with a dynamic
contact-line and the contact-line dynamics are of interest, whereas the former is
useful when the motion of the three-phase contact-line is well-known.
Figure 5.28 plots the oscillation frequency against the spreading parameter Λ
for the super-hemispherical base-state with α = 105◦ . Here the axisymmetric
k, l = 8, 0 mode shape is degenerate with respect to mode shapes with polar wavenumber k = 7 and azimuthal wave-numbers l = 3, 5, 7 at some finite value of the
spreading parameter. The most interesting comparison from figure 5.28 is between
the axisymmetric k, l = 8, 0 and sectoral k, l = 7, 7 mode shapes. Here the two
mode shapes have nearly identical frequencies in the pinned limit Λ → ∞, but the
transition to the natural regime Λ → 0 occurs at different values of the spreading
169
parameter. Moreover, the contact lines of the k, l = 7, 7 mode shape stay essentially
pinned for a larger range of Λ. This specific type of mode crossing behavior can be
used to explain one of the instabilities observed in drop-atomization experiments
(Vukasinovic et al., 2007). In these experiments, a sessile drop with pinned contactlines is placed on a vertically-driven vibrating plate and the driving frequency is
varied until resonance is observed. The frequency is then fixed and the resulting
shape is axisymmetric with pinned contact lines. Next, the forcing amplitude is
increased until the contact-line becomes de-pinned, which results in an azimuthal
disturbance generated along the contact-line. As the forcing amplitude is increased
further, the azimuthal instability propagates to the entire drop surface and finally
drop-atomization occurs. One can use figure 5.28 to relate the mode crossing
phenomenon to the generation of the azimuthal instability in the drop-atomization
experiments. Recall that the horizontal asymptotes of the curves from figure 5.28
are regions where the interface disturbance essentially satisfies either the natural
or pinned contact-line conditions. As the forcing amplitude increases, the mobility
of the contact-line increases or Λ decreases and the drop de-pins, albeit for an
instant. In relation to figure 5.28, the k, l = 8, 0 curve is traversed in the direction
of decreasing Λ for increasing forcing amplitude until this mode shape de-pins and
collapses onto the k, l = 7, 7 mode shape, which has a similar resonant frequency
and pinned contact-lines. Equivalently, as the contact-line de-pins an azimuthal
instability is necessarily generated to preserve the sessile drop’s preference for a
pinned contact-line. Although a quantitative comparison with these experiments
is not performed here, this type of mode crossing behavior is characteristic of the
de-pinning azimuthal instability.
Alternatively, mode crossings may be analyzed by fixing the disturbance class
and varying the volume via the static contact-angle. This has been done in fig-
170
(a)
(b)
Λk,l
23
9
21
19
40
k,lŠ5,1
k,lŠ5,3
k,lŠ5,5
k,lŠ6,0
k,lŠ6,2
Λk,l
12
k,lŠ6,0
k,lŠ6,2
k,lŠ6,4
k,lŠ5,3
k,lŠ5,1
k,lŠ7,7
6
50
60
70
80
Α
100
110
120
130
Α
140
Figure 5.29: Modal crossings controlled by base-state volume: frequency λk,l
as a function of static contact-angle α for (a) natural and (b)
pinned disturbances.
ure 5.29, which plots frequency λk,l against contact-angle α for a range of wavenumber pairs. Here the direction of mode crossings is controlled by category
of base-state, either sub-hemispherical or super-hemispherical. As the contactangle increases from 90◦ , high polar wave-number modes cross the low-polar wavenumber modes, as in figure 5.29(b). Likewise, figure 5.29(a) shows that decreasing
the contact-angle from 90◦ results in low wave-number modes crossing high wavenumber modes. In general, the number of possible mode crossings is controlled by
the magnitude of deviation from the hemispherical base-state.
Periodic table of mode shapes
As shown in figure 5.29, the ordering of mode shapes by their frequency can be
distorted for a range of contact angles. Detailed knowledge regarding the ‘filling’
of the frequency space is of fundamental interest and is of practical importance in
applications like the drop-atomization experiment, among others. A direct analogy
can be made between the filling of frequency space for the sessile drop and the
171
Figure 5.30: Spectral lines: (a) frequencies ωk,l , as they depend upon azimuthal wave-number l, against polar wave-number k and (b)
blow-up of polar wave-numbers k = 5, 6 with natural mode
shapes for α = 60◦ .
Figure 5.31: Spectral lines: (a) frequencies ωk,l , as they depend upon azimuthal wave-number l, against polar wave-number k and (b)
blow-up of polar wave-numbers k = 7, 8 with pinned mode
shapes for α = 120◦ .
172
filling of the periodic table by energy level. Consider the energy spectrum of
the hydrogen atom model in quantum mechanics, which consists of an electron
orbiting a nucleus under a spherically-symmetric potential. Here the energy levels
are degenerate with respect to angular momentum quantum number, much like the
sessile drop with hemispherical base-state is degenerate with respect to azimuthal
wave-number under a natural disturbance. Both problems can be formulated using
a variational approach and both degeneracies are attributed to the symmetry of
their respective systems. Similar to the degeneracy lifting of the sessile drop, a
number of symmetry-breaking mechanisms exist for the hydrogen atom model,
such as spin-orbit coupling, hyperfine splitting, etc. These mechanisms help to
explain the unique manner in which the periodic table is filled by energy level. For
example, the 4s orbital is filled before the 3d orbital, because the former state has
a lower energy than the latter. This peculiar filling order would not be possible if
the respective degeneracy were not broken. As shown in figures 5.30 & 5.31, the
frequency space of the sessile drop fills in a manner parallel to the filling of atomic
orbitals by energy level. Here the order in which resonant mode shapes were to
appear in a frequency sweep is identified by traversing the graph in the direction
of increasing frequency. Once again, the direction of splitting is independent of
the class of kinematic disturbance and controlled by the type of base-state, either
sub-hemispherical or super-hemispherical. The direction of frequency splitting is
indicated in figure 5.30 for a sub-hemispherical base-state α = 60◦ subject to a
natural disturbance and figure 5.31 for a super-hemispherical base-state α = 120◦
given a pinned disturbance. The magnitude of frequency splitting is controlled
by the static contact-angle α and contact-line conditions through the spreading
parameter Λ. Therefore, in making the analogy to the ‘periodic table of mode
shapes’, one must recognize that there exists a family of periodic tables that are
173
(a)
(b)
Figure 5.32: Periodic table of mode shapes: filling order for (a) subhemispherical drop (α = 60◦ ) subject to a natural disturbance and (b) super-hemispherical drop (α = 120◦ ) with pinned
contact-lines.
distinguished by α and Λ, but related by an Aufbau principle, which states that
the mode shapes are ‘filled’ in order of increasing frequency. For example, two such
periodic tables are shown in figure 5.32, which display the filling order for the set
of parameters used to generate the spectral plots of figures 5.30,5.31, respectively.
174
5.6
Concluding remarks
The dynamic stability of an inviscid sessile drop has been studied here. To that end,
the linearized hydrodynamic equations governing the motion of the drop interface
are formulated as a functional eigenvalue equation on linear operators. Here the
drop interface is subject to two distinct classes of disturbance, either i) kinematic
or ii) dynamic. A kinematic disturbance depends solely upon the geometry of
the three-phase contact-line, whereas a dynamic disturbance is related to dynamic
‘field’ quantities there. Two such kinematic disturbances are considered here; the
i) fixed contact-angle (natural) and ii) fixed contact-line (pinned). The dynamic
disturbance class uses a constitutive law that relates the ‘dynamic’ contact-angle
to the contact-line speed via a spreading parameter. Once the disturbance class
is established, the frequency spectrum for the sessile drop may be computed using
one of two equivalent formulations, deemed the forward and inverse problem. Both
problems are reduced to a set of algebraic equations using the variational procedure
of Rayleigh-Ritz. The kinematic disturbance class is handled most expediently by
solving the inverse problem, whereas an ad hoc solution method is used for the
contact-line speed condition. Here select results from the ‘natural’ disturbance
class are utilized and the structure of the second variation of surface energy is
exploited to show that contact-angle variation associated with the contact-line
speed condition is a purely dissipative process.
The computational results presented here are in excellent agreement with existing literature. To the extent that comparisons can be made, a few of the most
relevant points-of-contact will be summarized next. The vast majority involve
the hemispherical base-state (α = 90◦ ), because of the additional symmetry that
greatly reduces the complexity of the computations required. For example, the nat-
175
ural frequencies computed here, which are degenerate with respect to azimuthal
wave-number l, can be identified with (2.1) as the Rayleigh frequencies (Rayleigh,
1879). Likewise, Lyubimov et al. (2006) implement a linearized Hocking condition, which is related to the contact-line speed condition used here, to compute
the axisymmetric frequencies (l = 0), shown in figure 5.20 (d, e, f ), as a function
of a parameter inversely related to the spreading parameter Λ. A similar study
has been performed by Lyubimov et al. (2004) for non-trivial azimuthal wavenumbers l 6= 0 (c.f. figure 5.26). In contrast to the results mentioned above for the
hemispherical base-state, Basaran & DePaoli (1994) use a computational-based
approach to study the effect of base-state volume on the first axisymmetric mode
shape ([k, l] = [2, 0]). They assume pinned contact-lines and are able to handle
large-amplitude disturbances. Their computed frequencies, which do not vary significantly with the size of disturbance, are in excellent agreement with figure 5.6
(b), especially in the small-amplitude limit.
This problem is parameterized by azimuthal wave-number l, volume via the
static contact-angle α and the boundary conditions on the three-phase contact-line,
which can be controlled by the spreading parameter Λ. The spreading parameter
is a measure of the mobility of the contact-line and its asymptotic limits Λ → 0
and Λ → ∞ correspond to the natural and pinned disturbances classes, respectively. Equivalently, one can vary the spreading parameter and smoothly change
between the two kinematic boundary conditions. At finite values of the spreading
parameter, energy is dissipated according to dynamic contact-angle variation and
gives rise to damped oscillation amplitudes. As could be expected, the computed
decay rates are greatest for the higher wave-number mode shapes. However, it
is the low polar wave-number mode shapes that dissipate the most energy over
a complete oscillation cycle, counterintuitive to bulk viscous effects, where higher
176
wave-number modes dissipate more energy. This peculiarity is attributed to the
viscous dissipation mechanism of dynamic contact-angle variation controlled by
the contact-line mobility. It has been shown that the contact-line displacement is
largest for the lowest order modes, thus, these shapes dissipate the most energy
according to this constitutive law. To further validate this interpretation, it has
been shown that perturbations to sub-hemispherical α < 90◦ base-states dissipate
more energy than super-hemispherical α > 90◦ base-states, because of the greater
contact-line mobility of the former. Moreover, in the limit of the drop touching a
support plane on a generating line α → 180◦ , the contact-line is essentially immobile, no energy is dissipated and one cannot distinguish between the two kinematic
disturbances.
Although the majority of the motions of the sessile drop are oscillatory, there
does exist an unstable mode shape. Here a sessile drop subject to a natural disturbance exhibits instability for a range of contact angles 90◦ < α < 180◦ . This
instability is characterized by an advancing contact-line, which is accelerating relative to the corresponding receding contact-line, thereby displacing mass from one
side of the drop to the other. Under this disturbance class there is no such mechanism to resist this motion, therefore the drop is able to ‘walk’ along the support
surface. To suppress this ‘walking’ instability, one can force the contact lines to
be pinned, which precludes the translating motion. Alternatively, the contact-line
speed condition generates effective dissipation and can potentially suppress this
instability. That is, if the positive dissipation exactly balances the negative energy
difference between the perturbed state and the base-state, then the base-state is
neutrally stable (c.f. figure 5.24). A thorough understanding of this instability
potentially has many useful industrial applications, whereby translational motion
of droplets is desirable, such as the cleansing of microprocessor chips via fluid
177
droplets.
The special case of the hemispherical α = 90◦ sessile drop exhibits a frequency
degeneracy with respect to azimuthal wave-number l, when subjected to a natural
disturbance. This degeneracy is attributed to the additional configurational symmetry of the hemispherical base-state. Recall that the governing equations have
been derived from a variational principle and that Noether’s theorem states that
continuous symmetries imply degeneracies for Hamiltonian systems. Implicit in
Noether’s theorem is the assumption that the boundary conditions for the variational problem are the ‘natural’ ones. With regards to the sessile drop, the ‘natural’ boundary conditions are those that generate disturbances, which preserve
the static contact-angle or belong to the natural disturbance class. Thus, the azimuthal degeneracy of the hemispherical base-state could have been anticipated.
This degeneracy is lifted in two ways, either i) varying the boundary conditions
on the three-phase contact-line via the spreading parameter Λ or ii) adjusting the
volume of the base-state via the static contact-angle α. As shown in figure 5.19,
pinning the contact lines of the drop breaks this degeneracy. Here for a fixed polar
wave-number k, the frequencies with the highest azimuthal wave-number l can be
up to 10% greater than the mode shape with the lowest azimuthal wave-number
(c.f. figure 5.26). Likewise, volume effects break the degeneracy as in figure 5.18
for the natural disturbance class. This behavior persists for non-zero values of the
spreading parameter Λ. Furthermore, one can superimpose the two degeneracy
lifting mechanisms to create large frequency ranges for fixed polar wave-number,
much like the behavior shown in figure 5.27.
By controlling the volume of the base-state via the static contact-angle, it has
been shown that there exists the possibility for mode crossings (or the ordering of
178
mode shapes can be altered). Two types of mode crossing are observed. The first
is controlled by the spreading parameter Λ and is relevant to problems where the
contact-line dynamics are of interest. Typical mode crossing behavior of this type
is a potential explanation for a contact-line instability present in drop-atomization
experiments (Vukasinovic et al., 2007). Namely, a vertically-driven sessile drop
with pinned contact lines exhibits an instability, whereby its contact-line de-pins
for an instant and an azimuthal instability is generated. Although a quantitative
comparison is precluded, figure 5.28 illustrates this type of characteristic behavior. The second type of mode crossing behavior is related to the drop volume
through the static contact-angle and is shown in figure 5.29. For a fixed polar
wave-number k, the super-hemispherical base-states have oscillation frequencies
that increase as the azimuthal wave-number l increases. In contrast, frequencies
for sub-hemispherical base-states decrease with increasing azimuthal wave-number.
In both cases the relative difference in frequencies between the mode shapes with
smallest and largest azimuthal wave-number is controlled by the magnitude of deviation from the hemispherical base-state. Accordingly, as this deviation grows
the possibility of mode crossings greatly increases. Here if one were to perform a
frequency sweep, the order in which resonant mode shapes are excited is altered in
comparison to the classical modal ordering. The ‘filling’ of frequency space by these
broken-states is analogous to the filling of the periodic table by energy levels, where
the distinct ordering is a result of a number of symmetry-breaking mechanisms,
such as spin-orbit coupling, etc. With regards to the sessile drop, the symmetry
breaking mechanisms are the wetting conditions on the three-phase contact-line
and the base-state volume, which are controlled by the spreading parameter Λ
and static contact-angle α, respectively. For a prescribed set of parameter values
(Λ, α), one can construct the corresponding ‘periodic table of mode shapes’ from
179
the spectral data using an Aufbau principle, whereby the mode shapes are filled in
order of increasing frequency (c.f. figure 5.32).
180
CHAPTER 6
CAPILLARY INSTABILITIES OF THE STATIC RIVULET:
VARICOSE AND SINUOUS MODES
6.1
Introduction
A rivulet is a narrow stream of liquid flowing down a solid surface that shares an
interface, held by surface tension, with the surrounding gas. The most commonly
referred to example is the stream of water seen on automobile windshields. The
rivulet has the distinguishing feature of two distinct contact lines. The contactline is defined as the geometric curve that intersects the interface between two
immiscible fluids and a solid support. The equilibrium positions or the motion of
these contact-lines generates a number of geometric configurations. For example,
the straight rivulet has parallel contact lines, steady, fully-developed flow and a
cylindrical meniscus (Towell & Rothfeld, 1966). This base-state exhibits a number
of instabilities, such as drop formation from capillary break-up, rivulet meandering
and the development of surface waves (e.g. Schmuki & Laso, 1990).
A static rivulet is characterized by a constant mean curvature and the absence
of a velocity field. In the absence of a base-flow, the sole instability mechanism
is Plateau-Rayleigh break-up (capillary instability), whereby the interface evolves
into a series of individual droplets in an attempt to maximize its surface area. In a
series of experiments, Plateau (1863) showed that the liquid cylinder is unstable to
lengths longer than its base-state circumference. Plateau attempted to interpret
the final drop size from this limit, which was ultimately proved incorrect. Lord
Rayleigh (1879) corrected Plateau’s misinterpretation by computing the dispersion
relation from the governing hydrodynamic equations. Rayleigh was able to show
181
that the maximal growth rate and corresponding wave-number gave a characteristic time and size for drop formation, respectively. Davis (1980) computes static
stability bounds for the static rivulet under a number of contact-line conditions
but, unlike Plateau, derives his results from the hydrodynamic equations and corresponding disturbance energy-balance equation. One can directly compare Davis’
results to Brown & Scriven (1980), who use a variational approach to study the
cylindrical fillet protruding from an infinite slot for both constant pressure and
constant volume disturbances, because static stability is unaffected by the support
geometry. The agreement is perfect in the common limit of the constant volume
disturbance. A number of authors have studied cylindrical interfaces under other
wetting geometries. For example, Langbein (1990) studies the interior or exterior
wetting of a V-groove, Roy & Schwartz (1999) analyze a number of cross-sectional
containers; such as planar, V-groove, circular and elliptical, and Bostwick & Steen
(2010) consider the cylindrical-cup support.
A rivulet with an axial base flow is susceptible to kinematic-wave instabilities
characteristic of thin film flows–particularly so if the rivulet is relatively flat–in
addition to capillary instability. In a series of papers, Davis et al. study the
long wave-length instabilities on a rivulet corresponding to gravity-driven flow
down a vertical plane (Weiland & Davis, 1981; Young & Davis, 1987). They
report kinematic-wave instabilities for wide rivulets with immobile contact lines
and capillary instability of narrow rivulets. Although their analysis is able to
handle a wide range of effects and predict the varicose instability, they do not
capture rivulet meandering or the sinuous instability. For reference, the varicose
and sinuous modes are defined according to their symmetry or anti-symmetry
about the vertical mid-plane of a cross-section, respectively. Among other results,
Benilov (2009) uses the lubrication approximation to show that a shallow rivulet
182
with pinned contact lines that wets the under-side of a plate (pendant) is unstable
to a range of axial wave-numbers. In addition, an induced shear flow is seen to
stabilize this instability, which is a somewhat surprising result considering that
inertial effects tend to de-stabilize.
With respect to rivulet meandering, Culkin & Davis (1983) derive a stability
index to measure the stabilizing effects of surface tension and de-stabilizing effects
of inertia under dynamic wetting conditions. Unable to account for contact-angle
hysteresis, the stability index was only marginally effective at capturing the meandering instability mechanism observed in their experiments. Kim et al. (2004) use
a perturbation analysis to try to capture the meandering instability for a rivulet
base-state corresponding to plug flow. By balancing pressures at the contact-line,
they are able to derive a dispersion relation that depends upon the base-state geometry, a Weber number and wetting conditions on the contact-line. The response
of the fluid to a disturbed interface shape is not included in their analysis. Similarly, Grand-Piteira et al. (2006) derive a rivulet meandering criteria from a force
balance on the contact-line that incorporates contact-angle hysteresis, capillary
effects and inertia from a gravity-driven base flow. They find, among other observations, that the base-flow is also hysteretic and thus the shape of the meandering
rivulet varies only with increasing flow rate.
In this chapter, the linear stability of the static rivulet under a number of
contact-line conditions is considered. The first type of disturbance preserves the
static contact-angle (natural), while the second has a fixed contact-line (pinned).
The governing hydrodynamic equations for this inviscid, incompressible flow are
derived and then reduced to an eigenvalue problem by a normal mode analysis.
The problem is parameterized by axial wave-number k and static contact-angle
183
α (base-state volume). The integro-differential equation, governing the motion of
the interface, is formulated as a functional eigenvalue problem on linear operators,
which is then solved using a spectral method to deliver the dispersion relations.
Static stability is greatly influenced by the type of interface disturbance and can
be recovered from the hydrodynamic formulation by setting the growth rate to
zero.
Once the contact-line disturbance class is set, allowable solutions to the governing equations are decomposed further into the varicose and sinuous mode types.
That is, there are two mode types for both the natural and pinned disturbance
classes to yield four types of interface disturbance in total. With regards to stability, the varicose mode is the more unstable mode type and the pinned disturbance
is seen to be relatively stabilizing in comparison to the natural disturbance. Static
stability of the varicose mode is shown to agree with the results of Davis (1980).
The dispersion relations for the varicose mode exhibit behavior typical of capillary
instability. Specifically, there exists a fastest growing shape that is distinguished
by a non-trivial axial wave-number km . The sinuous mode with pinned contact
lines is stable to all static contact angles or base-state volumes. However, the sinuous mode does exhibit instability to the natural disturbance for the super-circular
(90◦ < α < 180◦ ) base-state. Unlike the varicose instability, this instability does
not exhibit the typical character of Plateau-Rayleigh break-up. For a range of contact angles, the fastest growing or critical disturbance is characterized by km = 0
or the disturbance is independent of the axial coordinate. This is a ’long-wave’
instability. Typical capillary instability persists outside this interval.
This chapter begins by defining the hydrodynamic equations that govern the
motion of the rivulet, whose interface has been given a small disturbance. A nor-
184
mal mode expansion reduces the governing equations to a functional eigenvalue
problem on linear operators. To compute the dispersion relations, a Rayleigh-Ritz
procedure is used to reduce the operator equation to a standard algebraic eigenvalue problem. The type of disturbance and volume of the base-state, controlled
by the static contact-angle α, are the relevant parameters in this problem and
can greatly influence the dispersion relations. The types of disturbance considered
here are defined both on the contact-line and according to their symmetry about
the vertical-mid plane. To conclude, the computational results are discussed and
concluding remarks are given.
6.2
Mathematical formulation
A cylindrical-cap interface (rivulet) held by uniform surface tension σ, on a planar
support, is a static equilibrium configuration. When gravitational effects are neglected, the static base-state is a circular arc in the x − y plane extended to infinity
in the axial (z) direction. A parametric representation of this equilibrium shape Γ
is given by
1
sin(s),
sin(α)
1
Y (s, z; α) =
(cos(s) − cos(α)) ,
sin(α)
X(s, z; α) = −
Z(s, z; α) = z,
(6.1a)
(6.1b)
(6.1c)
where s ∈ [−α, α] and z ∈ [−∞, ∞] are the arc-length and axial coordinates,
respectively. Here lengths have been scaled with respect to the base radius r. The
interface is given a small disturbance η(s, z, t) (c.f. figure 6.1) which generates
pressure gradients on the surface according to the Young-Laplace equation
p/σ = κ1 + κ2 ≡ 2H,
185
(6.2)
(a)
(b)
y
#
G
ΗHs,z,tL
r
Α
x
!
"
Figure 6.1: Definition sketch of the rivulet in (a) polar and (b) threedimensional perspective views.
and equivalently a capillary-driven flow. Accordingly, the circular arc base-state
has principal curvatures κ1 = sin α and κ2 = 0, a constant mean curvature 2H =
sin α, and is therefore in static equilibrium.
6.2.1
Field equations
The governing equations for this incompressible, irrotational flow are written via
a velocity potential v = ∇Ψ and pressure p;
∇2 Ψ = 0 [D]
p=ρ
∂Ψ
[D],
∂t
(6.3a)
(6.3b)
¡
¢
p/σ = −∆Γ η − κ21 + κ22 η [∂Df ].
(6.3c)
∂Ψ
∂η
=−
[∂Df ]
∂n
∂t
(6.3d)
∇Ψ · ŷ = 0 [∂Ds ]
(6.3e)
Here a fluid of density ρ occupies a domain
D ≡ {(x, y, z)| 0 ≤ x ≤ X(s, ϕ; α), 0 ≤ y ≤ Y (s, ϕ; α), −∞ ≤ z ≤ ∞}
186
(6.4)
that is bound by a free surface ∂Df and a planar surface-of-support ∂Ds ;
∂Df ≡ {(x, y, z) | x = X(s, z; α), y = Y (s, z; α), z = z},
(6.5a)
∂Ds ≡ {(x, y, z) | y = 0}.
(6.5b)
As required by incompressibility, the velocity potential satisfies Laplace’s equation
(6.3a) on the fluid domain and a no-penetration condition (6.3e) on the planar
surface-of-support. The pressure in the drop domain is expressed by the linearized
Bernoulli equation (6.3b). Similarly, the pressure evaluated on the free surface
is given by the linearized Young-Laplace equation (6.3c) with ∆Γ defined as the
Laplace-Beltrami operator or surface Laplacian (e.g. Myshkis et al., 1987). Lastly,
a kinematic condition (6.3d) relates the normal velocity of the fluid to the velocity
of the interface disturbance on the free surface.
6.2.2
Normal-mode reduction
Normal modes,
Ψ (x, t) = φ(x)eγt , η(s, z, t) = y(s, z)eγt
(6.6)
are used to reduce the hydrodynamic field equations (6.3) to
∇2 φ = 0 [D],
(6.7a)
∂φ
= 0 [∂Ds ],
∂n
¢ ∂φ
∂φ ¡ 2
−∆Γ
− κ1 + κ22
= λ2 φ [∂Df ],
∂n
∂n
Z
∂φ
= 0,
Γ ∂n
ργ 2 r3
λ2 ≡
.
σ
(6.7b)
(6.7c)
(6.7d)
(6.7e)
Here (6.7) is recognized as an eigenvalue problem in the scaled growth rate (6.7e)
with (6.7a-6.7c) following directly from (6.3). In addition, the auxiliary condi187
tion (6.7d) is required to ensure the linear perturbations are volume preserving
(incompressibility).
6.2.3
Operator formalism
Equation (6.7c) represents the balance of momentum on the interface and can be
simplified using the properties of the equilibrium surface,
·µ ¶
µ ¶¸
µ ¶
∂φ
∂φ
∂φ
2
+ sin α
+
= λ2 φ.
∂n zz
∂n ss
∂n
(6.8)
As the planar support is independent of the axial coordinate z, one may define
φ = φ eikz (with purposeful redundant notation) and Fourier transform (6.8) to
generate
µ
∂φ
∂n
¶00
¡
+ 1−k
2
¢
µ
∂φ
∂n
¶
=
λ2
φ.
sin2 α
(6.9)
The solution of this inhomogeneous differential equation is parameterized by axial
wave-number k and given by
∂φ
(s) = λ̂2
∂n
Z
α
G(s, τ ; k)φ(τ ) dτ ,
(6.10)
−α
where λ̂ ≡ λ/ sin α and G(s, τ ; k) is the Green’s function or fundamental solution
of the differential equation. Alternatively, one may view (6.10) as a functional
eigenvalue problem on linear operators,
M −1 [φ] = λ̂2 K −1 [φ] ,
(6.11)
with
M −1 [φ] ≡
∂φ
,
∂nZ
K −1 [φ] (s) ≡
(6.12a)
α
G(s, τ ; k)φ(τ ) dτ .
(6.12b)
−α
To compute the eigenvalue spectrum of (6.10) using the operator formalism, one
must construct the Green’s function or integral operator (6.12b).
188
(a)
(b)
y
y
G
r
G
Α
r
x
Α
x
Figure 6.2: Natural disturbances: polar view of the (a) varicose and (b) sinuous modes.
6.2.4
Contact-line conditions
Before one can construct the Green’s function, the boundary conditions on the
three-phase contact-line must be specified.
Natural
The first type preserves the static contact-angle α,
µ µ ¶
¶¯
∂ ∂φ
∂φ ¯¯
+ cos(α)
= 0,
∂s ∂n
∂n ¯s=±α
(6.13)
represents the linearization of the Young-Dupré equation (1.4), which is derived in
Appendix A, and is termed the natural disturbance (c.f. figure 6.2).
Pinned
The second class is shown in figure 6.3 and has immobile or ‘pinned’ contact-lines,
µ ¶¯
∂φ ¯¯
= 0.
(6.14)
∂n ¯s=±α
189
(a)
(b)
y
y
G
r
G
Α
r
x
Α
x
Figure 6.3: Pinned contact-line disturbances: polar view of the (a) varicose
and (b) sinuous modes.
Symmetry decomposition into varicose and sinuous modes
Here it has been assumed that the wetting conditions on both contact-lines s = ±α
are identical, which allows one to decompose the solutions of (6.7) according to
their symmetry about the vertical mid-plane. The varicose modes or even solutions
are symmetric and satisfy
µ
∂φ
∂n
¶0 ¯
¯
¯
¯
= 0,
(6.15)
s=0
while the anti-symmetric or odd solutions satisfy the following condition
µ
¶¯
∂φ ¯¯
= 0,
∂n ¯s=0
(6.16)
and are deemed the sinuous modes. For reference, typical mode types are shown
in figure 6.4.
190
(a)
(b)
!"#$
!"#$
Figure 6.4: Top view of typical three-dimensional (a) varicose and (b) sinuous
mode shapes.
6.2.5
Green’s function construction
The following representation of the Green’s function is particularly suited to the
symmetric decomposition,



 U (τ ;k)V (s;k)
G (s, τ ; k) =
W (s;k)


 U (s;k)V (τ ;k)
W (s;k)
0<s<τ <α
(6.17)
0 < τ < s < α.
Here U (left-hand) and V (right-hand) are the homogeneous solutions of (6.9) that
satisfy the boundary conditions on the vertical mid-plane s = 0 and the contactline s = α, respectively. Likewise, W is the Wronskian of the two solutions.
191
Right-hand solution
The right-hand solution V is independent of the symmetry about the mid-plane,
but does depend upon the wetting conditions on the three-phase contact-line. With
respect to the natural boundary conditions (6.13), the right hand solution is given
by
V n (s) = cos (βs) + A sin (βs)
(6.18)
β sin βα − cos α cos βα
,
β cos βα + cos α sin βα
(6.19)
with
A=
and
β≡
√
1 − k2.
(6.20)
Similarly, application of the pinned contact-line condition (6.14) generates the
following right hand solution,
V p (s) = sin β (s − α) .
(6.21)
To distinguish between disturbance classes, a superscript n will denote the natural solution, while a superscript p will refer to the pinned contact-line solutions.
Similarly, a subscript v or s will distinguish the varicose and sinuous modes, respectively.
Left-hand solution
With regards to the decomposition of solutions by symmetry, the left-hand solutions U are independent of the contact-line conditions. The varicose modes satisfy
(6.15) and have the left-hand solution
Uv (s) = cos (βs) ,
192
(6.22)
while the Wronskian for the natural Wvn and pinned Wvp disturbances are given by
Wvn (s) = Aβ,
(6.23a)
Wvp (s) = β cos(βα),
(6.23b)
respectively.
Similarly, the anti-symmetric sinuous modes satisfy (6.16) and have the following left-hand solution
Us (s) =
sin(βs)
,
β
(6.24)
with the Wronskians defined as follows
Wsn (s) = −1,
(6.25a)
Wsp (s) = sin(βα),
(6.25b)
for the natural and pinned disturbances, respectively.
6.2.6
Solution of operator equation
To compute the spectrum of (6.11), the operator equation is reduced to a truncated
set of linear algebraic equations using the variational procedure of Rayleigh-Ritz.
The method is sketched here, while a more thorough illustration is given in Segel
(1987). To begin, the eigenvalues λ of the operator equation
B [y] = λA [y]
(6.26)
are the stationary values of the functional
λ = min
(B[y], y)
, y ∈ S,
(A[y], y)
193
(6.27)
where S is a predetermined function space. A solution series,
y=
X
ci yi ,
(6.28)
i=1
constructed from functions yi ∈ S are applied to the functional (6.27) and minimized with respect to the coefficients ci to generate a set of algebraic equations
X
(bij − λaij ) cj = 0,
j=1
(6.29a)
Z
bij ≡
B[yi ] yj ,
(6.29b)
A[yi ] yj ,
(6.29c)
Z
aij ≡
from which the eigenvalues may be computed.
With regards to the problem considered here, a solution series
φ=
N
X
aj φj ,
(6.30)
j=1
constructed from properly chosen basis functions φj is applied to the operator
equation (6.11) and inner products are taken to generate the following algebraic
eigenvalue problem,
N ³
´
X
2
mij − λ̂ κij aj ,
j=1
(6.31a)
Z
α
∂φi
mij ≡
φj ds,
0 ∂n
Z αZ α
κij ≡
K −1 [φi ] φj dτ ds,
0
(6.31b)
(6.31c)
0
with
¶ µZ α
¶
U (s)
φj (s) ds
V (τ )φi (τ ) dτ
κij =
0
0 W (s)
Z α
Z s
Z α
Z s
V (s)
U (s)
+
φj (s)
U (τ )φi (τ ) dτ ds −
φj (s)
V (τ )φi (τ ) dτ ds.
0 W (s)
0
0 W (s)
0
µZ
α
(6.32)
194
Allowable solutions of (6.11) must satisfy the hydrodynamic equations and more
specifically, Laplace’s equation (6.7a) on the fluid domain and the no-penetration
condition (6.7b) on the surface-of-support. To satisfy these conditions, the following basis functions are used for the varicose modes,
(v)
φj = I2(j−1) (kr) cos(2(j − 1)θ),
(6.33)
and the sinuous modes,
(s)
φj = I2j−1 (kr) sin((2j − 1)θ),
(6.34)
where Il are modified Bessel functions (Arfken & Weber, 2001). Here the basis functions (6.33) & (6.34) are defined on the equilibrium surface through the
coordinate transformations,
r=
p
x2 (s; α) + y 2 (s; α),
x(s; α)
cos θ = p
,
x2 (s; α) + y 2 (s; α)
y(s; α)
,
sin θ = p
x2 (s; α) + y 2 (s; α)
(6.35a)
(6.35b)
(6.35c)
with x(s; α) and y(s; α) defined in (6.1). Similarly, the normal derivatives of the
basis functions are needed to compute the matrix elements of (6.31b),
∂
(I2j (kr) cos(2jθ)) =
∂n
k
[I2j+1 (kr) + I2j−1 (kr)] cos(2jθ) (− sin(s) cos(θ) + cos(s) sin(θ)) (6.36)
2
k
− [I2j−1 (kr) − I2j+ (kr)] sin(2jθ) (sin(s) sin(θ) + cos(s) cos(θ)) ,
2
∂
(I2j (kr) sin(2jθ)) =
∂n
k
[I2j+1 (kr) + I2j−1 (kr)] sin(2jθ) (− sin(s) cos(θ) + cos(s) sin(θ)) (6.37)
2
k
+ [I2j−1 (kr) − I2j+ (kr)] cos(2jθ) (sin(s) sin(θ) + cos(s) cos(θ)) .
2
Here the normal derivatives of the potential functions are given in mixed coordinates for efficiency in presentation and may be simplified further using the coordinate transformation (6.35).
195
(a)
Λ
0.5
(b)
2
Λ
0.10
45°
60°
90°
135°
0.4
0.3
0.06
0.04
0.1
0.02
0.5
1.0
1.5
105°
135°
150°
160°
170°
0.08
0.2
0.0
0.0
2
k
0.0
0.1
0.2
0.3
0.4
0.5
0.6
k
Figure 6.5: Dispersion relations: growth rate λ2 vs. axial wave-number k for
(a) varicose and (b) sinuous modes, subject to a natural disturbance.
6.3
Results
The eigenvalues of the matrix equation (6.31a), as they depend upon the axial
wave-number k, are computed using standard numerical techniques for both the
varicose and sinuous modes under either the natural (6.13) or pinned (6.14) dis(l)
turbance class. For a given eigenvalue λ̂2(l) with associated eigenvector aj , the
corresponding eigenfunction is the velocity potential
φ=
N
X
(l)
aj φj ,
(6.38)
j=1
which is related to the shape of the interface disturbance
y=
N
X
(l) ∂φj
aj
j=1
∂n
.
(6.39)
The unstable growth rates (λ2 > 0) and stable oscillation frequencies (λ2 < 0) are
computed using a truncation of N = 8 terms in the solution series (6.30). Here
recall that λ = (sin α)λ̂. Typical dispersion relations for the varicose and sinuous
modes are given in figure 6.5.
196
(a)
(b)
ks
2.5
ks
0.6
2.
0.5
Stable
0.4
1.5
0.3
1.
0.5
0
Ang-U,
0.2
Ang-U, Pin-S
45
Stable
90
135
Pin-S
0.1
Unstable
Α
180
0
45
90
135
Α
180
Figure 6.6: Static stability boundary ks against contact-angle α for (a) varicose and (b) sinuous modes for the natural (Ang) and pinned
(Pin) disturbances. Here stable (S) and unstable (U) regions are
noted.
6.3.1
Static stability
With regards to static stability ks , the varicose modes are more unstable than
the sinuous modes. That is, the varicose mode is unstable to a greater range
of axial wave-numbers, as shown in figure 6.6. In addition, for a fixed mode
type a pinned contact-line disturbance is always stabilizing when compared to the
natural disturbance. Static stability results for the varicose mode are summarized
in figure 6.6(a) and can be directly compared to those computed by Davis (1980).
The agreement is excellent. Despite the fact that the varicose mode is the dominant
instability, the sinuous mode does exhibit instability to natural disturbances when
the cylindrical base-state is super-circular α > 90◦ . However, pinning the contact
lines stabilizes the sinuous mode for all contact angles (and, equivalently, volumes)
(c.f. figure 6.6(b)).
197
(a)
(b)
Λ2m
km
1.5
0.5
Ang
Ang
0.4
Pin
Pin
1.
0.3
0.2
0.5
0.1
0
45
90
135
180
Α
0
45
90
135
180
Α
Figure 6.7: Fastest growing varicose mode: (a) wave-number km and (b)
growth rate λ2m against contact-angle α for natural (Ang) and
pinned (Pin) disturbances.
6.3.2
Critical disturbance
A characteristic feature of the Plateau-Rayleigh instability is the existence of a
fastest growing mode-shape distinguished by its axial wave-number km and growth
rate λ2m . For example, the critical disturbance to the unconstrained Rayleigh jet
(liquid cylinder) has a maximum growth rate λ2m = 0.343 and wave-number km =
0.69. As shown in figure 6.7, the critical wave-number and maximum growth rate
are modified according to the disturbance class and static contact-angle, but the
behavior characteristic of Plateau-Rayleigh break-up persists for all of the varicose
modes. Figure 6.8(a, b) displays typical varicose instability mode shapes for both
disturbance classes. In the limit of a cylinder touching a planar support along a
generating line (α → 180◦ ), the stability results for the natural and pinned contactline disturbance are indistinguishable (c.f. figures 6.6(a),6.7). More specifically,
the contact-line is essentially immobile for the natural disturbance in this limit.
The sinuous mode exhibits behavior unlike that of the varicose mode and uncharacteristic of the Plateau-Rayleigh instability. For example, figure 6.9 shows
198
(a)
(b)
(c)
(d)
!"#$
Figure 6.8: Typical instability mode shapes. The varicose mode for the (a)
natural and (b) pinned contact-line disturbance. The natural
sinuous mode with a (c) polar (k = 0) and (d) typical axial
(k 6= 0) disturbance.
that for a range of contact-angle 90◦ < α < 150◦ the wave-number of maximum
growth is km = 0; that is, the critical disturbance is a polar one. Figure 6.8(c)
plots a typical instability mode shape from this regime. Physically, this instability
is characterized by an advancing contact-line, which is accelerating relative to the
receding contact-line, thereby causing the cylindrical interface to ‘walk’ horizontally along the surface-of-support in the direction normal to the cylindrical axis.
Outside this interval, typical capillary instability or Plateau-Rayleigh break-up
occurs, as in figure 6.8(d).
Lastly, one can compare the effects of the type of support-surface. For example,
figure 6.10 plots the maximum instability growth rate for the varicose modes of a
cylindrical interface with pinned contact lines and in contact with either a planar
or cylindrical-cup support (Bostwick & Steen, 2010) against contact-angle. Since
199
(a)
(b)
km
0.4
Λ2m
0.1
0.08
0.3
0.06
0.2
0.04
0.1
90
0.02
120
150
180
Α
90
120
150
Α
180
Figure 6.9: Fastest growing sinuous mode: (a) wave-number km and (b)
growth rate λ2m against contact-angle α for the natural disturbance.
Λ2m
0.12
Cylindrical
Planar
0.09
0.06
0.03
90
120
150
Α
180
Figure 6.10: Comparison between the varicose mode with pinned contactlines contacting either a cylindrical or planar support: maximum growth rate λ2m against contact-angle α.
it depends solely upon the free surface shape, static stability is unaffected by the
support geometry and is therefore identical for these two configurations. However,
the maximal growth rate for the interface constrained by a cylindrical-cup support
(see figure 4.1), reported by Bostwick & Steen (2010), is always larger than that
for the corresponding interface in contact with a planar support.
200
6.4
Concluding Remarks
A number of authors have investigated the series of instabilities exhibited by fluid
rivulets. It is common practice to use the lubrication approximation for film flow
and asymptotics in the long wave-length limit to study the dynamics of the rivulet.
Young & Davis (1987) use these approximations to study the varicose modes of
the rivulet with a unidirectional gravity-driven base flow down a vertical plane.
In the absence of a base flow, the rivulet is susceptible to capillary instability or
Plateau-Rayleigh break-up, which can be greatly affected by the wetting conditions
on the three-phase contact-line. For example, Davis (1980) computes the static
stability bounds for the static rivulet under a number of contact-line conditions.
In this chapter, the hydrodynamic equations are derived and then solved to give
the dispersion relations for the static rivulet subject to the contact-line conditions
used by Davis (1980). Thus, this work is intended to extend the work of Davis
(1980) in a manner similar to how Lord Rayleigh (1879) extended the work of
Plateau (1863) on the stability of liquid cylinders.
The dispersion relations for the static rivulet are computed from a functional
eigenvalue equation using a Rayleigh-Ritz variational procedure. One major benefit associated with formulating the problem in a functional analysis setting is
that interchanging boundary conditions on the contact-line and/or mode types
is straightforward. Static stability of the varicose modes agrees well with Davis
(1980), who does not need to distinguish between varicose and sinuous modes,
because the varicose modes are the more unstable of the two mode types. This is
confirmed by the computations performed here.
The sinuous mode exhibits instability to the super-circular base-state that is
subject to a fixed contact-angle (natural) disturbance. This instability does not
201
share the typical characteristics of Plateau-Rayleigh break-up. More specifically,
for a range of contact angles, the critical or fastest growing disturbance is distinguished by an axial wave-number km = 0 which, unlike Plateau-Rayleigh break-up,
are ‘long-wave instabilities’. The instability mode shape is characterized by an
advancing contact-line, which is accelerating relative to its receding contact-line.
Thus, fluid is displaced from one side of the rivulet to the other in what amounts
to translational center-of-mass motion in the horizontal direction normal to the
cylindrical axis (c.f. figure 6.8(c)). Outside this interval, typical capillary instability occurs, as shown in figure 6.8(d). Cataloging the sinuous instability may
be important in understanding the phenomenon of rivulet meandering. From this
analysis, rivulet meandering can only occur when the contact-lines are mobile.
With respect to the varicose modes, the rivulet is unconditionally unstable
to natural disturbances and unstable to the super-circular base-states when the
contact-lines are pinned. Accordingly, the growth rates of the critical disturbance
are always larger for the natural disturbances, because they are less restrictive
than a disturbance with pinned contact-lines. Finally, to compare the effects of
the constraint geometry, the maximum growth rates for the cylindrical interface
in contact with a planar support (rivulet) and a cylindrical-cup support (Bostwick
& Steen, 2010) are compared. Here, static stability of the two configurations is
coincident, but a larger critical growth rate occurs for the cylindrical-cup support.
202
CHAPTER 7
STABILITY BOUNDS FOR THE CATENOID
7.1
Introduction
One of the classic problems in the calculus of variations involves constructing
surfaces of minimal area for a given set of boundary conditions. The catenoid
is an example of one of these minimal surfaces, which are characterized by their
mean curvature being zero. The connection between the mathematical description
of these equilibrium surfaces and the theory of capillarity is well established and
originated with the works of Young (1805) and Laplace (1806), who formulated
the basic principles of surface tension. Namely, by assuming a fluid interface held
by constant surface tension has a configurational energy proportional to its surface
area, one can show a capillary surface is in static equilibrium if it has a constant
mean curvature, and equivalently a constant pressure. Accordingly, static capillary
surfaces are governed by the nonlinear Young-Laplace equation (an Euler-Lagrange
equation). The resulting family of capillary surfaces is parameterized by mean
curvature. Given a particular equilibrium surface, one would like to know under
what conditions this configuration might be realized physically. To address this
issue, one must introduce the notion of stability.
‘Static stability’ generally refers to the potential energy associated with a specific configuration, whereby one can infer relative stability by comparing the energies of two adjacent configurations. The state with lower energy is stable relative
to the one with higher energy. Additionally, one can say that the equilibrium
configuration in question is a local minima of energy, or is locally stable, if the
comparison of adjacent configurations is exhaustive. For example, Plateau (1863,
203
1873) has used this notion of stability to derive the well-known Plateau limit: a
cylindrical interface is unstable if its axial length is longer than the base-state
circumference. One should note that the aforementioned notion of stability has
implicitly assumed that the equilibrium configurations in question have been subject to the same ‘class’ of disturbance. In the theory of capillary surfaces, two
disturbance classes are of particular interest; those which preserve either i) volume
or ii) pressure of the equilibrium configuration.
In general, two calculations are involved when determining the static stability
of a given equilibrium; determining the i) equilibrium surface and then its corresponding ii) stability. The latter is normally more complicated than the former.
According to the calculus of variations, to deduce stability one must show the second variation is positive. One way is to prove i) Legendre’s condition and ii) the
absence of a conjugate point or negative eigenvalue of Jacobi’s equation (Bolza,
1904). The stability calculation can be complicated further through the introduction of a conservation of volume constraint, an auxiliary condition needed for
incompressible fluids. Using a conjugate point criteria, Howe (1887) has worked
directly with the second variation to generate stability results for the zero-gravity,
axisymmetric capillary surface under the constant-volume constraint. These results for the axisymmetric liquid bridge have been summarized by Gillette & Dyson
(1971) in the zero-gravity limit.
To partially circumvent the calculation of the second variation, one may utilize
the tools of bifurcation theory and constrained variational principles. The methodology is sketched here and expanded upon further in Appendix D. To begin, one
may introduce a Lagrange multiplier in order to embed the volume in the surfacearea functional and calculate families of equilibria from the augmented functional.
204
This is an iso-perimetric problem. As shown by Maddocks (1987) for an arbitrary
Hilbert space, stability limits are always located at the ‘turning points’ of a preferred bifurcation diagram derived from the augmented functional. This method
projects the second variation onto a constrained function space and uses an index
theory to deliver the desired result. To use the theory, some information regarding
the unconstrained second variation is needed. Hence, the stability calculation may
be greatly simplified, but not eliminated. This theory has been applied to the
axisymmetric liquid bridge by Lowry & Steen (1995).
Alternatively, one may work indirectly with the second variation by exploiting
its structure and using elementary results from the calculus of variations to obtain
static stability bounds for the family of catenoids, as well as the axisymmetric
liquid bridge. A similar method has been used by Davis (1980) for the static
rivulet under a variety of contact-line conditions. The technique developed in this
chapter is termed the ‘bounding’ method. To validate the use of the bounding
method, the static stability limit for the catenoid with fixed contact-lines (pinned)
is computed and compared with known results, Erle et al. (1970). Next, static
stability bounds are derived for the class of disturbances that preserves the static
contact-angle. There are apparently no existing results regarding the stability of
the catenoid subject to the fixed contact-angle (natural) disturbance. Lastly, the
linearized hydrodynamic problem is solved and dispersion relations are computed
in order to compare with the stability limit obtained by the bounding method.
The stability computed from the hydrodynamic approach is referred to as ‘dynamic’ stability. Equilibrium configurations are dynamically stable if starting from
a bounded set of initial disturbances, the evolution of the disturbance stays within
the ‘neighborhood’ of the equilibrium for all time. Static stability can be recov-
205
ered from the linearized hydrodynamic equations by setting the growth rate to
zero. Thus, one can directly compare the stability limits from the hydrodynamic
approach and the bounding method, which will serve as another validation of the
proposed method. In addition, the computed linear growth rates give physical
insight into the development of this instability.
The stability of the catenoid, subject to a variety of disturbances, is considered
in this Chapter. First, a method is proposed to compute the static stability limit of
the family of catenoids without explicitly calculating the second variation. Here the
extreme-value theorem and some elementary results from the calculus of variations
allow one to derive the result, which agrees well with the previous literature in the
appropriate limit. To validate the static results computed from this method, the
linearized hydrodynamic equations are formulated and growth rates are computed,
from which one can compare directly with the static stability limit. The agreement
is excellent. Finally, the bounding method is applied to the general axisymmetric
liquid bridge to derive the corresponding static stability bound. Here information
with regards to the equilibrium surface is the only information required to be input
to the stability criteria.
7.2
Static stability
The catenoid is a surface-of-revolution, a solution of the Young-Laplace equation,
can be defined parametrically as
³s´
cos ϕ,
c
³s´
Y (s, ϕ; c) = c cosh
sin ϕ,
c
X(s, ϕ; c) = c cosh
Z(s; c) = s,
206
(7.1a)
(7.1b)
(7.1c)
z
R=1
L2
x
L2
Α
Figure 7.1: Definition sketch for the catenoid.
using arc-length s ∈ [−S, S] and azimuthal angle ϕ ∈ [0, 2π] as generalized surface
coordinates (c.f. figure 7.1). This family of surfaces, as they depend upon the
parameter c, have principal curvatures
³ ´
1
2 s
κ1 = −κ2 = sech
c
c
(7.2)
and may be characterized by their mean curvature H ≡ κ1 + κ2 = 0.
As the catenoid is a solution of the Euler-Lagrange equations, stability is determined by giving the equilibrium surface a perturbation y(s) and solving the
eigenvalue problem associated with the second variation
¡
¢
−∆Γ y(s) − κ21 + κ22 y(s) = λy(s).
(7.3)
The sign of the eigenvalue λ determines the stability of the equilibrium surface.
Here the Laplace-Beltrami operator or surface Laplacian,
1 ∂
∆Γ y = √
g ∂uα
µ
207
√
gg
αβ
∂y
∂uβ
¶
,
(7.4)
is defined on the equilibrium surface through the

2
0
 cosh (s/c)
gαβ ≡ xα · xβ = 
0
c2 cosh2 (s/c)
surface metric


4
2
 , g = c cosh (s/c).
(7.5)
Finally, the eigenvalue problem (7.3) is augmented with a boundary condition at
the three-phase contact-line and may or may not satisfy the volume conservation
constraint
Z
y(s)ds = 0.
(7.6)
Γ
That is, the disturbance class to which the interface is subject may consist of either
constant volume or constant pressure disturbances.
As an alternative to constructing an explicit solution to the boundary value
problem (7.3), one can work directly with the second variation. By using the
definition of the surface metric (7.5), the integral form of the second variation
(7.3) can be written as a functional equation
Z S
³s´
y 2 ds
I=λ
cosh2
c
−S
with
Z
S
I≡−
−S
µ
³ ´ ¶
2
2 s
y + 2 sech
y y ds.
c
c
00
(7.7)
(7.8)
Using this formulation proves advantageous, because the structure of (7.7) allows
one to pose a sufficient condition for stability.
Theorem 2 The catenoid is stable to infinitesimal disturbances when the functional (7.8) is positive.
To prove this theorem, note that the right hand side of (7.7) is positive for all functions y. The value of the functional I is dependent upon the boundary conditions
at the three-phase contact-line and will be analyzed separately.
208
7.2.1
Pinned contact-line disturbance
In this section the contact-line is assumed to be immobile or pinned,
y(−1/2) = y(1/2) = 0,
(7.9)
and solutions to (7.7) are sought in both the constant pressure and constant volume
disturbance classes. Here lengths have been scaled such that S = 1/2 and the
family of catenoids is parameterized by c.
To derive a stability criteria, begin by integrating functional (7.8) by parts and
using contact-line conditions (7.9) to give
Z
1/2
µ
³ ´ ¶
2
2 s
y 2 ds,
y − 2 sech
c
c
02
I=
−1/2
(7.10)
which is a quadratic form that can be bounded on the interval s ∈ [−1/2, 1/2],
Z
1/2
−1/2
µ
2
y − 2 sech2
c
µ
02
1
2c
¶
¶
y
2
Z
1/2
ds ≥ I ≥
−1/2
µ
¶
2 2
y − 2 y ds,
c
02
(7.11)
using the extreme-value theorem. It follows from (7.11) that if
Z
µ
1/2
¶
2 2
y − 2 y ds > 0,
c
02
Imin ≡
−1/2
(7.12)
the stability criteria of Theorem 2 is satisfied.
To compute the value of Imin , as it depends upon c, the following elementary
result from the calculus-of-variations, often called the ‘Poincare inequality’, is used
Z
Z
1/2
02
y ds ≥ ξ
1/2
2
−1/2
y 2 ds.
(7.13)
−1/2
This result is applied to the functional (7.12) to give
µ
Imin =
2
ξ − 2
c
¶Z
2
209
1/2
−1/2
y 2 ds,
(7.14)
and equivalently the following stability criteria
c2 >
2
.
ξ2
(7.15)
Here the positive number ξ 2 , defined in (7.13), is the smallest eigenvalue calculated
from the boundary value problem
y 00 + ξ 2 y = 0,
y(−1/2) = y(1/2) = 0.
(7.16)
As mentioned earlier, two classes of solution to (7.16) are considered. The first
class has a constant-pressure and the solution is given by
yp (s) = cos(πs),
ξp2 = π 2 ,
(7.17)
while the second class has constant-volume,
yv (s) = sin(2πs), ξv2 = (2π)2 .
(7.18)
Note that (7.18) satisfies (7.6) and therefore preserves volume. The smallest eigenvalue ξ 2 is the necessary input to the stability criteria (7.15) and is used to compute
the critical parameter values
√
cp =
2
1
, cv = √ ,
π
2π
(7.19)
for constant pressure and constant volume disturbances, respectively.
Results
The family of catenoids can be represented in the length-volume space by defining
the slenderness
Λ≡
L
1
=
2R
c cosh (1/2c)
210
(7.20)
V
1
0.8
0.6
Stable
Vol-S, Press-U
Unstable
0.4
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
L
Figure 7.2: Stability diagram for the pinned contact-line disturbance class
showing stable (solid line), unstable (dashed line) and conditionally stable (dotted) catenoids. Here the conditionally stable
catenoid is stable (S) to constant-volume (Vol) and unstable (U)
to constant-pressure (Press) disturbances.
and relative volume
R 1/2
V ≡
−1/2
cosh2 (s/c) ds
cosh2 (1/2c)
.
(7.21)
The computed critical slenderness for constant pressure Λp = 0.66 and constant
volume Λv = 0.47 disturbances are summarized in the stability diagram of figure 7.2 (pinned catenoids). The stability of the catenoid with pinned contact lines
has been reported by Erle et al. (1970), among others. However, the bounding
method is new and has been shown to be quite satisfactory, as witnessed by the
0.01% error in the critical slenderness. In addition, one may infer that the instability is local in nature, because the extreme value theorem is inherently local in that
it isolates the location of the maximal sum of squares of the principal curvatures.
211
(a)
S
(b)
c
1.0
0.6
0.5
0.4
0.3
0.2
0.1
0.8
0.6
0.4
0.2
Π
4
Π
2
Α
Π
4
Π
2
Α
Figure 7.3: Scaling of catenoid by contact-angle: parameters (a) S and (b) c
against contact-angle α.
7.2.2
Natural disturbance
The second class of disturbances preserves the static contact-angle α, in accordance
with the Young-Dupré equation (1.4), and is termed the natural disturbance. Here
lengths are scaled by the base radius R and the arc-length coordinate is defined as
s ∈ [−S, S], such that
S≡
2 sinh−1 (cot α)
¡
¢
cosh sinh−1 (cot α)
(7.22)
c=
1
¢
cosh sinh−1 (cot α)
(7.23)
and
¡
are parameterized by the contact-angle α (c.f. figure 7.3). Equivalently, the family of catenoids can be represented in the length-volume space by defining the
slenderness
Λ≡
L
sinh−1 (cot α)
¡
¢
=
2R
cosh sinh−1 (cot α)
(7.24)
and volume
"
¡
¢#
sinh 2 sinh−1 (cot α)
1
¡
¢ 1+
V ≡
,
2 cosh2 sinh−1 (cot α)
sinh−1 (cot α)
212
(7.25)
which is scaled with the volume of a cylinder. To preserve the static contactangle, allowable disturbances must satisfy the following kinematic conditions on
the three-phase contact-line,
(−y 0 + k(α)y) |s=−S = 0, (y 0 + k(α)y) |s=S = 0
with
cot α
k(α) ≡
sech2
c(α)
µ
S
2c(α)
(7.26)
¶
.
(7.27)
To formulate the stability criteria for the natural disturbance, transform the
functional (7.8) into a quadratic form
¶ ¶
µ
Z Sµ
£
¤
2
s
2
02
I=
y − 2
sech
y 2 ds − k(α) y 2 (S) + y 2 (−S)
c (α)
c(α)
−S
(7.28)
using boundary conditions (7.26) and integration by parts. Similar to what was
done previously, the quadratic form (7.28) is bounded from below
¶
Z Sµ
£
¤
2 2
02
I ≥ Imin =
y − 2
y ds − k(α) y 2 (S) + y 2 (−S)
c (α)
−S
(7.29)
using the extreme value theorem and stability again follows from Theorem 2. An
inequality from the calculus of variations,
Z S
Z
£ 2
¤
02
2
y ds − k(α) y (S) + y (−S) ≥ ζ
−S
S
y 2 ds,
(7.30)
−S
is used to recast the critical functional (7.29) as
¶Z S
µ
2
y 2 ds
Imin = ζ(α) − 2
c (α)
−S
(7.31)
with stability assured if
ζ(α) −
2
c2 (α)
> 0.
(7.32)
The Euler-Lagrange equations associated with the functional (7.29) give rise to
the boundary value problem,
y 00 + ζy = 0, (−y 0 + k(α)y) |−S = 0, (y 0 + k(α)y) |S = 0,
213
(7.33)
whose smallest eigenvalue ζ(α) is required in the stability criteria (7.32). As before,
the constant pressure solution of (7.33) is expressed as
p
p
p
ζp
yp (s) = cos( ζp s),
ζp tan
= k(α)
2
(7.34)
while the constant volume solution satisfies (7.6) and is given by
yv (s) = sin(
p
p
ζv s),
ζv cot
√
ζv
= −k(α).
2
(7.35)
Finally, to compute the critical contact-angle α from (7.32), the smallest eigenvalue
ζ(α) from (7.34) and (7.35) is needed.
Results
Computations show the constant pressure disturbance is unstable for all contact
angles and the constant volume disturbance is stable for contact angles α > 21.6◦ ,
or a critical slenderness Λ = 0.61. These results are summarized in the natural
disturbance stability diagram of figure 7.4. The constant pressure stability result
is not surprising, considering the contact-line is free to move along the surface of
support with prescribed contact-angle. For example, as the contact-line moves,
volume is allowed to leave the domain, resulting in the collapse of the catenoid,
because energy decreases with decreasing surface area. This phenomenon is not
present when volume is conserved as when the contact-line is pinned. Similar to
the pinned disturbance, the conservation of volume constraint acts like a restoring
force to the contact-line motion, much like the spring force resists the inertial
motion in a simple harmonic oscillator.
214
V
1
0.8
0.6
Vol-S, Press-U
Unstable
0.4
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
L
Figure 7.4: Stability diagram for the catenoid subject to the natural disturbance class showing unstable (dashed line) and conditionally
stable (solid line) regions. The conditionally stable catenoid is
stable (S) to constant-volume (Vol) and unstable (U) to constantpressure (Press) disturbances.
V
1
0.8
0.6
Stable
Pin-S, Ang-U
Unstable
0.4
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
L
Figure 7.5: Constant-volume stability diagram showing stable (solid line),
unstable (dashed line) and conditionally stable (dotted line)
catenoids. The conditionally stable catenoid is unstable (U) to
natural (Ang) and stable (S) to pinned (Pin) disturbances.
215
V
1
0.8
0.6
Pin-S, Ang-U
Unstable
0.4
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
L
Figure 7.6: Constant-pressure stability diagram showing unstable (dashed
line) and conditionally stable (solid line) catenoids. The conditionally stable catenoid is unstable (U) to natural (Ang) and
stable (S) to pinned (Pin) disturbances.
7.2.3
Remarks
One may compare the effect of the disturbance class to which the catenoid is
subject, either natural or pinned. As evident from figures 7.5 & 7.6, pinning
the contact-line is shown to be stabilizing with respect to both constant-volume
and constant-pressure disturbances, respectively. To clarify, when viewed as a
variational problem constraints tend to stabilize by restricting the class of allowable
solutions, as is the case with the pinned contact-line.
As stated earlier, the bounding technique localizes the variational problem by
isolating the location of the maximal sum of squares of principal curvatures, which
occurs at s = 0 for the catenoid. This implies that the instability is local, which
is somewhat surprising considering that the disturbance classes are distinguished
by their contact-line conditions, a large distance from s = 0. Nevertheless, the
stability limit for the pinned disturbance is reconciled with the previous literature
216
to a more than acceptable degree of accuracy for this type of local approximation.
To further validate this method, the stability limit for the natural disturbance will
be calculated using the hydrodynamic approach. This approach is used, because
the computed growth rates are physically significant and the static results are
recovered by setting growth rate to zero.
7.3
Dynamic stability
The linearized hydrodynamic equations, governing the motion of the catenoidal
interface, are formulated in this section. To compare with the static stability
results obtained by the bounding method, solutions are sought in the space of
functions which preserve static contact-angle α and conserve volume. To analyze
this class of disturbance, the contact-angle parametrization of the catenoid (7.22)
and (7.23) is used. Likewise, lengths are scaled with the base radius R, as shown
in figure 7.1.
7.3.1
Hydrodynamic formulation
The domain (c.f. figure 7.1)
D ≡ {(x, z)| 0 ≤ x ≤ X(s; α), −S ≤ z ≤ S}
(7.36)
of this inviscid fluid is bounded by a free surface ∂Df and two parallel end-plates
∂Ds ,
∂Df ≡ {(x, z) | x = X(s; α), z = Z(s; α)},
(7.37a)
∂Ds ≡ {(x, z) | z = ±S}.
(7.37b)
217
Recall that the total arc-length S implicitly defines the axial length L = 2S, as in
(7.22). The irrotational velocity field of this incompressible fluid is described by a
velocity potential, v = ∇Ψ, which satisfies Laplace’s equation
∇2 Ψ = 0 [D]
(7.38)
on the fluid domain and a no-penetration condition on the end-plates,
∇Ψ · ẑ = 0 [∂Ds ].
(7.39)
The catenoidal interface is displaced by a small axisymmetric perturbation η(s, t),
whose velocity must be kinematically compatible with the normal velocity on the
interface,
∂η
∂Ψ
=−
[∂Df ].
∂n
∂t
(7.40)
An interface given such a perturbation generates pressure gradients
¡
¢
p/σ = −∆Γ η − κ21 + κ22 η [∂Df ]
(7.41)
balanced by the Bernoulli pressure
p=ρ
∂Ψ
[D],
∂t
(7.42)
evaluated at the fluid interface, as required by continuity. Equations (7.38)-(7.42)
are the time-dependent field equations governing the motion of an incompressible,
inviscid fluid, whose catenoidal interface is subject to a small axisymmetric disturbance. These equations are reduced to an eigenvalue problem by a normal mode
analysis.
7.3.2
Normal-mode reduction
Normal modes
η(s, t) = y(s)eiωt , Ψ(x, t) = φ(x)eiωt ,
218
(7.43)
are used to reduce the hydrodynamic equations (7.38)-(7.42) to an eigenvalue problem
∇2 φ = 0 [D],
(7.44a)
∂φ
= 0 [∂Ds ],
∂n
¢ ∂φ
∂φ ¡ 2
−∆Γ
− κ1 + κ22
= λ2 φ [∂Df ],
∂n
∂n
Z
∂φ
= 0,
Γ ∂n
ρω 2 R3
λ2 ≡
.
σ
(7.44b)
(7.44c)
(7.44d)
(7.44e)
Equations (7.44a)-(7.44c) follow directly, whereas (7.44d) is necessary to enforce
incompressibility or volume conservation. One should note that (7.44c) represents
the balance of inertial and capillary pressures at the interface and is equivalent
to the eigenvalue problem associated with the second variation (7.3). In contrast,
the eigenvalue λ2 , which measures either the unstable growth rate or frequency of
oscillation, may be interpreted physically. This follows from the fact that (7.44 a −
d) were derived from the hydrodynamic equations. Moreover, static stability is
obtained by setting the eigenvalue to zero.
Operator formalism
Equation (7.44c) is an operator equation that can be formulated using one of two
equivalent approaches. The first uses the normal interface velocity as the basis
function and is referred to as the forward problem,
· ¸
· ¸
∂φ
∂φ
2
K
=λ M
,
∂n
∂n
(7.45)
while the second has the velocity potential as the relevant basis function and is
known as the inverse problem,
M −1 [φ] = λ2 K −1 [φ].
219
(7.46)
Stationary values of both functionals necessarily satisfy Laplace’s equation (7.44a)
and conserve volume according to (7.44d), but the forward problem requires a sufficiently general solution to the following Neumann type boundary value problem,
¯
∂ψ ¯¯
∇ ψ = 0,
= yk (s),
∂n ¯Γ
2
(7.47)
which is not analytically tractable for this geometry. One approach would be to
use a more computationally intensive approach. Alternatively, one may treat
M −1 [φ] ≡ cosh2
³ s ´ ∂φ
(7.48)
c ∂n
as the differential operator and compute the integral operator, inverse to K. This
formulation is called the inverse problem and will be used to compute the spectrum
of (7.44).
7.3.3
Solution of operator equation
The eigenvalue spectrum of (7.44) is computed by a Rayleigh-Ritz procedure. This
procedure reduces the inverse operator formulation (7.46) of the governing equation
to a set of algebraic equations or a standard eigenvalue problem. To begin, one
must construct the inverse operator K −1 .
Inverse-operator construction
The differential operator
2
K [y] ≡ y + 2
sech2
c (α)
00
220
µ
s
c(α)
¶
y
(7.49)
is defined on the equilibrium surface and subject to the following boundary conditions on the contact-line,
¯
¯
(−y 0 + χy) ¯s=−S = 0, (y 0 + χy) ¯s=S = 0,
µ
¶
1
s
2
sech
χ(s) = cot(α)
,
c(α)
c(α)
(7.50)
a necessary condition to ensure interface disturbances preserve the static contactangle. The inverse operator
Z
K
−1
S
[y] ≡
G(s, q)y(q) dq
(7.51)
−S
is related to the fundamental solution of the corresponding differential operator,
or Green’s function,
G(s, q) =



v1 (q)v2 (s) + 1 w(s, q) −S < q < s < S
2


v1 (s)v2 (q) + 1 w(s, q) −S < s < q < S.
2
(7.52)
The functions v1 and v2 satisfy
¶
µ
2
s
2
00
vk + 2
sech
vk = 0, v1 (0) = 0, v10 (0) = 1, v2 (0) = 1, v20 (0) = 0 (7.53)
c (α)
c(α)
and belong to the kernel of K, while
µ ¶
µ ¶
τ1
τ2
w(s, q) =
v2 (s)v2 (q) −
v1 (s)v1 (q) − v1 (s)v2 (q) − v1 (q)v2 (s)
τ2
τ1
1
+ F (s)F (q)
τ3
(7.54)
with
τ1 = v10 (S) + χ(S)v1 (S), τ2 = v20 (S) + χ(S)v2 (S),
(7.55a)
·µ ¶
¸
τ2
F (s) = [v2 (s)ψ1 (s) − v1 (s)ψ2 (s)] +
ψ2 (S) − ψ1 (S) v2 (s),(7.55b)
τ1
Z s
ψk (s) =
vk (q) dq,
(7.55c)
0
Z S
τ3 = −
F (s) ds.
(7.55d)
0
221
Rayleigh-Ritz reduction
The eigenvalue spectrum may be computed by reducing the operator equation
(7.46) to a set of linear algebraic equations using a Rayleigh-Ritz procedure with
the following basis functions,
1
φk (r, z) = ck [Φk (r, z) − dk ] , dk ≡
S
Z
S
Φk |Γ ds,
(7.56)
0
which are chosen to satisfy the no-penetration condition (7.44b) identically. The
basis functions are scaled such that their average value on the equilibrium line is
zero, a sufficient condition to ensure volume is conserved, as required by the assumption of incompressibility. Owing to the symmetry of the boundary conditions
(7.50) on the three-phase contact-line, the eigenvalue problem may be decomposed
into its odd/even extension using the pertinent basis functions,
³
³
πz ´
πr ´
(o)
sin (k − 1/2)
,
Φk (r, z) = I0 (k − 1/2)
L
L
³ πr ´
³ πz ´
(e)
Φk (r, z) = I0 k
cos k
.
L
L
(7.57a)
(7.57b)
Here I0 is the modified Bessel function of the first kind and L = 2S with S given by
(7.22). A superscript “o, e” will be used to distinguish the odd and even solutions,
respectively.
The necessary input to the Rayleigh-Ritz procedure is a solution series
φ=
N
X
ak φk ,
(7.58)
k=1
constructed from the basis functions (7.56), which are used to reduce the functional
eigenvalue problem (7.46) to the following set of linear algebraic equations,
X¡
¢
mik − λ2 κik ak = 0,
k=1
mik ≡
Z
(7.59a)
Z
M
−1
[φi ]φk ds, κik ≡
222
K −1 [φi ]φk ds.
(7.59b)
The corresponding matrix elements in the odd/even extensions are expressed as
Z S
∂φi
(o)
(o)
mik = mki = 2ci ck
cosh2 (s/c(α))
(s)φk (s) ds,
(7.60a)
∂n
0
¸
·µ ¶
τ2
(o)
(o)
κik = κki = −2ci ck
ζi ζk − ζi ηk − ζk ηi + ²ik + ²ki , (7.60b)
τ1
and
Z
(e)
mik
(e)
=
S
∂φi
= 2ci ck
cosh2 (s/c(α))
(s)φk (s)ds,
∂n
0
·µ ¶
µ ¶
¸
τ1
1
= 2ci ck
ηi ηk +
ξi ξk − ²ik − ²ki ,
τ2
τ3
(e)
mki
(7.61a)
(e)
(7.61b)
κik = κki
respectively. Here the following definitions have been used
Z S
ζk ≡
v1 (s)φk (s)ds,
0
Z S
ηk ≡
v2 (s)φk (s)ds,
0
Z S
Z s
²ik ≡
v1 (s)φi (s)
v2 (q)φk (q)dq ds,
0
0
Z S
F (s)φk (s)ds.
ξk ≡ −
(7.62a)
(7.62b)
(7.62c)
(7.62d)
0
Finally, the scale factor ck , from (7.60) and (7.61), is chosen such that the coordinate functions are of order one on the equilibrium surface,
µZ S
¶−1/2
1
∂φk
2
ck = √
cosh (s/c(α))
(s)φk (s)ds
.
∂n
2
0
7.3.4
(7.63)
Results
The eigenvalues, as they depend on the contact-angle α, are computed from (7.59)
using N = 8 terms in the solution series (7.58). Computations exhibit relative error
for the first three eigenvalues of 0.1% for this truncation. Given an eigenvalue λ2j
(j)
and eigenvector ak , the instability or oscillatory mode shape is given by
µ
¶
N
X
∂φk
(j)
(j)
y (s) =
ak ck
(s).
∂n
k=1
223
(7.64)
(a)
(b)
Λ21
Λ21
1
1
0.45
0.5
0.55
0.6
0.65
L
15
-1
-1
-2
-2
-3
-3
-4
-4
20
25
30
Α
Figure 7.7: Unstable growth rate λ21 < 0 vs. (a) slenderness Λ and (b)
contact-angle α for the catenoid given a natural disturbance.
y1
1
0.5
-1
0.5
-0.5
1
s
-0.5
-1
Figure 7.8: Typical instability mode shape for the natural disturbance (α =
20◦ ).
The primary goal in formulating the hydrodynamic equations was to compare
with the static stability results from the bounding method. Computations reveal
the j = 1 mode shape is unstable λ2 < 0 for a range of contact angles α < 21.6◦ ,
as shown in figure 7.7(b). Likewise, the critical slenderness Λ = 0.61, seen in figure 7.7(a), is in excellent agreement with the static stability results of the previous
section, obtained by the bounding method. For reference, a typical unstable mode
shape is shown in figure 7.8 using a scaled arc-length coordinate. Additionally,
224
(a)
(b)
y2
Λ22
100
0.5
80
60
-1
40
20
0.5
-0.5
1
s
-0.5
10
15
20
25
30
Α
-1
Figure 7.9: Oscillation mode (n = 2): (a) Frequency λ22 > 0 vs. contactangle α and (b) sample mode shape for α = 25◦ .
(a)
(b)
y3
Λ23
550
1
450
0.5
350
-1
250
0.5
-0.5
1
s
-0.5
150
10
15
20
25
30
Α
-1
Figure 7.10: Oscillation mode (n = 3): (a) Frequency λ23 > 0 vs. contactangle α and (b) sample mode shape for α = 20◦ .
the first mode shape, important in applications as it is the first to be excited, will
exhibit oscillations with characteristic frequency for contact angles greater than
the critical value.
The higher order mode shapes are readily computed and may be of interest.
Here, oscillatory motion λ2 > 0 persists for mode numbers j ≥ 2. These results are
summarized in figures 7.9 & 7.10, which plot frequency as a function of contactangle, as well as typical j = 2 and j = 3 mode shapes, respectively.
225
7.4
Static stability of the axisymmetric liquid bridge
In view of the satisfactory results of the bounding technique, as it applies to the
catenoid, one could ask if such a stability criteria could be formed for any liquid
bridge or surface-of-revolution. To begin, a parametric description of an axisymmetric surface-of-revolution is given by
x(s, ϕ) = %(s) cos ϕ, y(s, ϕ) = %(s) sin ϕ, z(s, θ) = ψ(s),
(7.65)
using arc-length s ∈ [−S, S] and azimuthal angle ϕ ∈ [0, 2π] as surface coordinates.
The surface metric associated with this surface

02
02
0
£ αβ ¤−1
 % +ψ
gαβ = g
≡ xα · xβ = 
0
%2
of revolution is expressed as

¡ 02
¢

2
02
(7.66)
; g = % % + ψ ,
and can be used in conjunction with the definition of the Laplace-Beltrami operator
(7.4) to generate the second variation


´0
³p
Z S
Z S
gθθ /gss
¡ 2
¢ 2
 00
0
2
´ y y − gss κ1 + κ2 y  ds = λ
gss y 2 ds,
−y y − ³p
−S
−S
gθθ /gss
(7.67)
associated with this solution of the Euler-Lagrange equations. Here, the following
definitions have been used
gss = %02 + ψ 02 , gθθ = %2 .
(7.68)
If one assumes that the surface has pinned contact lines,
y(−S) = y(S) = 0,
(7.69)
the second variation (7.67) can be recast as
Z
S
I=λ
gss y 2 ds,
−S
226
(7.70)
where I is a quadratic form given by
# #
"
µ³
Z S"
´0 ³p
´¶0
p
¡
¢
1
2
y 2 ds.
I≡
(y 0 ) − gss κ21 + κ22 −
gθθ /gss /
gθθ /gss
2
−S
(7.71)
As done previously, this functional can be bounded
Z Sh
i
2
I ≥ Imin ≡
(y 0 ) − Ky 2 ds
(7.72)
−S
using the extreme value theorem, where the constant K,
"
µ 0 ¶2 #
¡ 2
¢
%
1
1
K ≡ max gss κ1 + κ22 + κ1 κ2 +
,
2
2 %
(7.73)
is simply the maximum value of the functional on the domain s ∈ [−S, S]. As
before, one may use a result from the calculus of variations (7.13) to recast (7.72)
as
¡
2
Imin = ξ − K
¢
Z
S
y 2 ds.
(7.74)
−S
The stability criteria can now be stated for the constant pressure,
³ π ´2
,
2S
K<
(7.75)
and constant volume disturbance
K<
³ π ´2
S
,
(7.76)
respectively.
7.4.1
Plateau-Rayleigh instability
To illustrate the effectiveness of the bounding technique, consider the classic
Plateau-Rayleigh instability of a cylindrical liquid bridge. The cylindrical interface
has principal curvatures κ1 = 1, κ2 = 0 and the following parametric representation
% = 1, ψ = s, gss = 1.
227
(7.77)
Here, lengths have been scaled by the radius of the cylinder and the family of
cylinders is uniquely described by its axial length L = 2S. The functional (7.73)
has constant value K = 1 for the cylindrical interface and is required to compute
the stability limit. According to the stability criteria (7.75) and (7.76), the cylinder
is stable to constant pressure disturbances if
³ π ´2
1<
L
,
(7.78)
and constant volume disturbances provided
µ
1<
2π
L
¶2
,
(7.79)
or for lengths smaller than the base-state circumference. Hence, the stability window from the bounding technique and the classic Plateau limit are coincident,
further validating this approach.
7.4.2
Remarks
In general, stability calculations involve solving a boundary value problem associated with the second variation or computing conjugate points, which prove to be
cumbersome in all but the simplest geometries. This difficulty, as it pertains to
the boundary value problem, is circumvented by utilizing the bounding method
proposed here. Specifically, the stability criteria, (7.75) and (7.76), are uniquely
determined by the geometric properties of the equilibrium shape through the constant K (7.73) and total arc-length 2S.
228
7.5
Concluding remarks
The linear stability of the catenoidal interface has been analyzed using both a
static and dynamic approach. Static stability is determined by the sign of the
second variation of surface energy, while dynamic stability can be inferred from
the computed growth rates, or solutions of the governing hydrodynamic equations.
Recall, the static limit can be recovered from the dynamics by setting the growth
rate to zero. In both approaches, ‘allowable’ solutions are sought in one of two
disturbance classes, distinguished by their respective boundary conditions on the
three-phase contact-line. The first is termed ‘natural’ and preserves the static
contact-angle, in accordance with the Young-Dupré equation, while the second
has a fixed/immobile contact-line and is referred to as ‘pinned’. The disturbance
class is enlarged further in the static approach to include both constant-pressure
and constant-volume disturbances. As could be expected, stability can be greatly
influenced by the disturbance class considered.
Using the static approach, a sufficient condition for stability of the catenoid is
derived by a bounding method. This method exploits the structure of the second
variation, which can be manipulated into a quadratic form and bounded using
the extreme value theorem. The lower bound of the quadratic form is a critical
functional from which stability may be deduced. Finally, by exploiting elementary
results from the calculus of variations, a criteria for static stability is stated for the
natural and pinned disturbances. The static stability limit for the catenoid with
pinned contact lines is a well-known result and can be used to validate the bounding method, which is in excellent agreement with previous literature (Erle et al.,
1970). Using the same approach, it has been shown that natural disturbances are
destabilizing when compared to the pinned contact-line disturbance class. Specif-
229
ically, a catenoid subject to a natural disturbance is shown to be unstable for all
contact angles when pressure is held constant. Alternatively, one may stabilize
the catenoid to this disturbance class by fixing the volume. Here the family of
catenoids with contact-angle α > 21.6◦ are stabilized. Lastly, one may infer that
the instability of the catenoid is a local instability, because the bounding method
relies on the inherently local extreme value theorem.
It has been shown, by application of the bounding method, that one need
not construct a general solution to the boundary value problem associated with
the second variation to generate a sufficient condition for stability. To reconcile
the stability limit obtained by the bounding method, the hydrodynamic equations
are derived and linear growth rates are computed. The dynamic approach gives
rise to an integro-differential equation, governing the interface deflection, which is
shown to be structurally equivalent to the second variation of the static approach.
Growth rates are then computed by formulating the governing boundary value
problem as a functional eigenvalue equation on linear operators. Computations
show the static and dynamic stability limits are coincident, further validating the
bounding method. That is, a catenoid subject to a natural disturbance exhibits
oscillatory motion provided α > 21.6◦ . Unlike the static approach, the dynamic
approach gives rise to instability growth rates and oscillation frequencies that may
be interpreted physically.
Finally, to illustrate the utility of the bounding method, a sufficient condition
for static stability of the axisymmetric liquid bridge with pinned contact lines is
derived. As with the catenoid, one only needs information regarding the geometric
properties of the equilibrium surface to generate the relevant stability criteria. A
limiting case of the liquid bridge is the cylindrical interface, whose stability is ex-
230
pressed by the well-known Plateau limit and recovered using the bounding method.
Although the bounding method is clearly an approximation, it has produced quite
satisfactory results and requires very little analysis or computation, as is the case
for most stability calculations.
231
CHAPTER 8
FUTURE WORK
To conclude this thesis, a number of final remarks will be given with regards
to open questions that stem from this work.
8.1
‘Walking’ instability
One of the more interesting and perhaps unexpected results presented here is the
‘walking’ instability of the [1, 1] mode shape, which has been related to the drop’s
horizontal center-of-mass motion. There remains a number of questions to be answered regarding this instability, most of which are related to the small amplitude
assumption, linearization of the governing equations and more specifically the constitutive law that relates the contact-angle to contact-line speed α = f (uCL ) there.
As shown in figure 5.24, the super-hemispherical base-state is unstable for a range
of values of the spreading parameter Λ–the spreading parameter is an artifact from
the linearization of the constitutive law. Interpretation of the analysis presented
here depends upon how appropriate a model for experiment that constitutive law
is. In addition, how relevant is its linearization, shown in figure 5.5, to the dynamics at the contact-line of any particular experimental system? As stated earlier,
contact-angle hysteresis is observed in experiment but not allowed in the analysis presented here, because of the small-amplitude assumption. One extension of
this analysis would be to include moderate-amplitude disturbances, which would
allow one to study the effect of the weakly-nonlinear terms and also contact-angle
hysteresis on the walking instability.
Another related issue is the possible coupling of oscillatory modes to the walking
232
instability. For example, Noblin et al. (2009) report ratchet-like motion of vibrated
drops, whereby the translational motion or the walking instability is enhanced by a
‘pumping’ motion, which appears to be related to the drop’s vertical center-of-mass
motion. An obvious question is why does this motion enhance the [1, 1] instability,
if indeed, it does so? Finally, can any mode shape be coupled with the [1, 1] mode
to enhance droplet motion and which one generates the greatest enhancement?
8.2
Forced oscillations
One would like to compare the theoretical predictions given in this thesis against
experimental results. To compare, one must answer the following question: to what
extent can the assumptions made in this analysis be realized experimentally? In
general, drop motion is induced by some type of external forcing, which may or may
not have a large influence on the dynamics. For example, consider the two following
experimental set-ups used to generate drop motion. How might they relate to the
harmonic oscillator structure of the governing equations derived here? The first
uses acoustic excitation to induce drop motion. The harmonic oscillator analogue
to this experimental set-up is pure mechanical forcing, whereby the drop response
is most pronounced at the natural frequencies. Alternatively, one might excite
the drop by a mechanically-vibrated plate, which generates pressure gradients on
the drop surface and is more characteristic of parametric excitation, as occurs in
Faraday waves (Faraday, 1831). Although this experimental set-up differs greatly
from the theoretical analysis, one might use some of the characteristic features
of the parametrically-excited oscillator to relate the theory to experiment. More
specifically, the defining feature of parametric resonance is the period-doubling
phenomenon, whereby droplet motion is induced when the forcing frequency is
233
equal to twice the natural frequency. As before, the natural frequency data is
available from the current analysis and may serve as a guide in determining if
parametric resonance occurs. Finally, it is possible to simultaneously excite two
distinct modes if the natural frequencies for the two mode shapes satisfy ωA /2 =
ωB = ωF . Here a forcing frequency ωF can generate a state of parametric resonance
in mode A and pure resonance in mode B.
It has been suggested that results from this thesis may be used to interpret
one of the instabilities seen in the experiments by Vukasinovic et al. (2007), where
a de-pinning event at the contact-line triggers an azimuthal instability there. As
shown in figure 5.28, a modal crossing occurs at some finite value of the spreading
parameter Λ, which is implicitly related to the forcing amplitude via the contactangle variation. In their experiments, Vukasinovic et al. (2007) show that this
transition always takes an axisymmetric into a non-axisymmetric shape. However,
it seems plausible that such a modal crossing may occur between two distinct
non-axisymmetric shapes.
8.3
Contact-line speed condition
Computations show that implementation of the contact-line speed condition (5.30)
generates effective dissipation controlled by the spreading parameter Λ (c.f. figures 5.20-5.23). The effective dissipation results from the particular form of the
contact-line constitutive law and scales differently than bulk viscous effects. It
would be interesting to conduct experiments to verify this scaling. If successful,
the experiments would characterize the spreading parameter and also validate the
assumption of this particular contact-line constitutive law.
234
APPENDIX A
LINEARIZATION OF YOUNG-DUPRÉ EQUATION
The Young-Dupré equation represents the balance of forces at the three-phase
contact-line of a fluid interface and relates the vectors normal to the free surface
(n) and surface-of-support (n1 ) to the contact-angle α. Similarly, the variation
(linearization) of the Young-Dupré equation generates the relevant boundary condition on the three-phase contact-line that preserves the static contact-angle α. To
derive this condition, consider the perturbed equilibrium surface shown in figure 1.2
and decompose the variation δx to the equilibrium surface into its perpendicular
δ⊥ x and parallel δk x components,
δx = δ⊥ x + δk x = y n + δk x.
(A.1)
Here the linear surface perturbation y has been defined as
δ⊥ x ≡ y n,
(A.2)
or with respect to the variation of the equilibrium surface in the direction of the
surface normal n. The variation of the Young-Dupré equation gives
δ (n · n1 ) = − (sin α) α̂
(A.3)
or a balance equation which relates the geometry of the disturbed interface to the
deviation in static contact-angle α̂. One can use the decomposition (A.1) to write
the geometric (kinematic) variation of (A.3) as
¢
¡
¢
¡
δ (n · n1 ) = δ⊥ n + δk n · n1 + n · δ⊥ n1 + δk n1 ,
235
(A.4)
which can be further simplified with following identities;
∂y
e,
∂s
∂n
δk n =
(e · δx) ,
∂s
δ⊥ n = −
(A.5a)
(A.5b)
δ⊥ n1 = 0,
δk n1 =
∂n1
(e1 · δx) .
∂s1
(A.5c)
(A.5d)
Here (A.5a) follows directly from the definition of the surface perturbation (A.2),
(A.5b) & (A.5d) are standard differentials and (A.5c) represents the variation
normal to the support surface, which is trivial because the motion of fluid interface
there is strictly in the direction e1 tangent to the support. In addition, s and s1 are
arc-length coordinates defined on the equilibrium surface and surface-of-support,
respectively.
The Frenet-Serret equations in the plane,
∂n
= −ke,
∂s
(A.6)
relates the directional change in the vector normal n with respect to its arc-length
coordinate s to the normal curvature k of that curve in the tangential direction e
(see Kreyszig, 1991). This equation applies to both the free surface (s, n, e) and
surface-of-support (s1 , n1 , e1 ) and allows one to reduce (A.5b) & (A.5d) to
∂n
(e · δx) · n1 = −k (e · n1 ) (e · δx) ,
∂s
∂n1
(e1 · δx) · n = −k̄ (e1 · n) (e1 · δx) ,
∂s1
(A.7a)
(A.7b)
where the normal curvatures of the free and support surfaces have been defined as
k and k̄, respectively. The vectors n, n1 , e, e1 are coplanar and are related by the
236
following vector identities;
e · δx = (n · n1 ) e1 · δx = (cos α) e1 · δx,
(A.8a)
n · δx = (n · e1 ) e1 · δx + (n · n1 ) n1 · δx,
(A.8b)
e1 · n = −e · n1 = sin α.
(A.8c)
Finally, one uses (A.7) and the vector identities (A.8) on (A.4) to generate the
linearized Young-Dupré equation
¶
µ
∂y
k̄
+ k cot α −
y = −α̂.
∂s
sin α
(A.9)
If the variation in contact-angle α̂ = 0, then (A.9) boundary condition on the
three-phase contact-line that ensures the linear surface disturbance y preserves the
static contact-angle.
237
APPENDIX B
MODIFIED BOUNDARY CONDITION CALCULATIONS FOR THE
IMMISCIBLE VISCOUS DROP
B.1
Modified boundary conditions
Here the details required to determine Tni and Tne from the modified boundary
conditions (3.66) are given. For efficiency in presentation, define
h
i
h
i
1/2
∗
en ≡ H(1)
Jen ≡ Jn+3/2 (γ ∗ /²i )1/2 ; H
(µγ
/²
)
.
i
n−1/2
(B.1)
To begin, use the definition of the velocity field (3.21) on (3.66a) to give
·
Tni
¸
·
¸
¸
·
1
1
2n + 1
e
+ Tn
= −γdn
.
en
n(n + 1)
Jen
H
(B.2)
Similarly, equations (3.17a),(3.21) & (3.57) used on (3.66b) give
· ½ µ
¶
µ
¶
¾¸
2 (n + 2)
2
2 (n − 1)
2
i
e
µe T n 1 +
+ γdn
} − µi {Tn 1 −
+ γdn
en
n+1
n
H
Jen
·
¸
(B.3)
1
(1)
i 2
(1)
× [1 − Γ(x)] Pn (x) = µi −Tn
− γdn
Γ(x)Pn (x)
n
Jen
To isolate Tni and Tne , recognize that the sum on n is implicit and both sides of
(1)
(B.3) may be expanded as a series in Pn (x) to give:
¶
¸
· µ
¶
¸
·
µ
Ln
2
2
i
e
T n µi
− µi 1 −
(1 − Ln ) + Tn µe 1 +
(1 − Ln )
en
Jen
Jen
H
· µ
¶
¸
n−1
n+2
Ln
= γdn 2 µi
− µe
(1 − Ln ) − µi
.
n
n+1
n
(B.4)
Equations (B.2) and (B.4) are solved to give Tni and Tne as functions of dn , or the
interface perturbation y.
238
B.2
Immiscible drop operator equation
To derive the operator equation for an immiscible drop, begin by using the scalings
(3.69) to recast (3.68) in the form
¶
¸
X½·µ
1
1
∗2
∗
ρ
+
γ + 2 (n − 1) (n + 2) (µ − 1) γ ²i dn + n (n + 2) µγ ∗ ²i Tne
n+1 n
n=1
¾
£¡
¢
¤
∗
i
− (n − 1) (n + 1) γ ²i Tn = − 1 − x2 yxx − 2xyx + 2y ,
(B.5)
where Tni and Tne are determined by solving equations (B.2,B.4) to give
³
´
£
¡
¢
¤ 1
2
2n+1
n+2
Ln
µ 1 + He (1 − Ln ) n(n+1)
+ 2 (1 − Ln ) n−1
−
µ
−
²
i
e
n
n+1
n
H
h n
³
´
i
h ³
´
i n dn (B.6)
Tni =
1
²i LJen − 1 − Je2 (1 − Ln ) − Je1 µ 1 + He2 (1 − Ln )
e
H
n
n
n
n
n
and
h³
Tne =
1−
2
Jen
´
1
en
H
i
£ ¡
¢
¤ 1
2n+1
n+2
Ln
(1 − Ln ) − ²i LJen n(n+1)
− 2 n−1
−
µ
(1
−
L
)
−
²
n
i
n
n+1
n
Jen
n
h
³
´
i
h ³
´
i
dn .
²i LJen − 1 − Je2 (1 − Ln ) − Je1 µ 1 + He2 (1 − Ln )
n
n
n
n
(B.7)
Equations (B.6) & (B.7) may be substituted into (B.5) to produce an integrodifferential operator equation for an immiscible drop with material parameters
ρ, µ and ²i .
239
APPENDIX C
ROTATIONAL WAVE SOLUTION OF THE VISCOUS DROP
UNDER SPHERICAL-BELT CONSTRAINT
Previously, the assumption was made that the vorticity could be written in the form
(3.18) using a properly chosen vector potential B. This assumption gave rise to
solutions with non-trivial radial velocities at the interface or the shape oscillations.
To derive the rotational wave solutions, one introduces a vector potential A that
generates a radial component of vorticity,
ω = ∇ × ∇ × A, A = A(r, θ)er .
(C.1)
This class of solution does not have a radial component to its velocity field, but
rather tangential velocities. As with the other field quantities, A(r, θ) can be
expanded as
A(r, θ) =
X
Sn (r)Pn (cos θ).
(C.2)
n=1
Substitution of the vector potential (C.1) into the vorticity equation (3.11c) gives
µ d2 Sn
µ n (n + 1)
+
γS
−
Sn = 0,
n
ρ dr2
ρ
r2
(C.3)
whose general solution is written as
(1)
³ r ´1/2
³ r ´1/2
Hn+1/2 (z e )
Jn+1/2 (z i )
i
i
e
e
Sn (R)
, Sn (r) =
Sn (R) (1)
Sn (r) =
R
Jn+1/2 (Z i )
R
H
(Z e )
(C.4)
n+1/2
with
µ
z
i,e
ρi,e
≡r γ
µi,e
¶1/2
, Z
i,e
µ
¶1/2
ρi,e
≡R γ
.
µi,e
(C.5)
The remaining unknowns Sni (R), Sne (R) are determined from the shear boundary
conditions
h
vϕi = vϕe
i
e
τrϕ
= τrϕ
i
∂D1f , ∂D2f ,
h
i
∂D1f , ∂D2f ,
vϕi = vϕe = 0
240
[∂Ds ] .
(C.6a)
(C.6b)
(C.6c)
As before, using the indicator function these conditions may be transformed into
a uniform set of boundary conditions,
vϕi |r=R = vϕe |r=R ,
¡ i
¢
e
τrϕ − τrϕ
|r=R [1 − Γ (x, ζ1 , ζ2 )] = CΓ (x, ζ1 , ζ2 ) vϕi |r=R ,
(C.7a)
(C.7b)
valid on the entire interface. The following relations,
¸
·
1 dSn dPn
1 dPn
2
Sn (r), τrϕ = µ 2 Sn (r) −
,
vϕ = −
r dθ
r
r dr
dθ
(C.8)
are applied to the modified boundary conditions (C.7) to generate the characteristic
equation,
¡ ¢
µHn+1/2 (X e ) − Jn+1/2 X i = (n + 2) (µ − 1) + (1 − µ)
Ln
,
1 − Ln
(C.9)
that determines the growth rate of the rotational waves. Here the following definitions have been used,
X i ≡ (γ ∗ /²i )1/2 ,
(C.10a)
X e ≡ (µγ ∗ /²i )1/2 ,
¶µ
µZ 1
¶
¡ (1) ¢2
2n + 1 (n − 1)!
Pn (x) Γ(x, ζ1 , ζ2 ) dx
Ln ≡
.
2 (n + 1)!
−1
(C.10b)
241
(C.10c)
APPENDIX D
CONSTRAINED VARIATIONAL PRINCIPLE
Consider the variational problem of minimizing a given functional H(z), but
subject to a finite number of subsidiary conditions
Ci (z) = ki ,
i = 1, · · · , p.
(D.1)
Here z can be a vector in Rn or a function in a Hilbert space H. In either representation, one generally introduces Lagrange multipliers µi and defines the augmented
Lagrangian
F (z; µi ) = H (z) −
p
X
µi Ci (z)
(D.2)
i=1
to enforce the auxiliary conditions (D.1). An equilibrium or critical point z ∗ of the
Lagrangian F necessarily has a vanishing first variation,
∗
∗
∇F (z ; µi ) = ∇H (z ) −
p
X
µi ∇Ci (z ∗ ) = 0.
(D.3)
i=1
Here ∇ denotes the appropriate derivative, gradient in Rn or the variational derivative in a function space. The structure of the constraints (D.1) reduces the first
order conditions to
∇F (z ∗ ; µi ) = ∇H (z ∗ ) = 0
(D.4)
and one recognizes that the Euler-Lagrange equations for the functionals H, F are
identical, but a ‘constrained’ extremal necessarily lies in the constraint manifold,
M = {z | Ci (z) = ki , i = 1, · · · , p}.
(D.5)
With regards to stability, one can ask under what conditions is an extremal of H
a constrained minimum (Maddocks, 1985) or can the functionals Ci and data ki
be chosen to make the extremal a constrained minima (Hestenes, 1951). When
242
referring to the stability of a constrained extremal, the weaker characterization
is a conditional stability, which treats perturbations to the extremal that lie in
the constraint manifold (D.5). A sharper definition of stability would involve
perturbations to the data ki , which define the constraint manifold itself. Both
notions of stability involve analyzing the quadratic form of the second variation
on a restricted sub-space or the constrained space.
The stronger definition of stability involves the augmented Lagrangian (D.2):
Lemma 1 An extremal z ∗ of H is a constrained minimum if the following second
order conditions are met,
¡
¢
h, ∇2 F (z ∗ )h > 0 {∀ h 6= 0 s.t. ∇Ci (z ∗ ) · h = 0,
i = 1, · · · , p}.
(D.6)
As opposed to restricting to perturbations h that lie in the tangent space of the
constraint manifold, an equivalent definition of stability involves the use of projection operators P ,
¡
¢
h, P t ∇2 F (z ∗ )P h > 0, ∀ h 6= 0.
(D.7)
Unlike the stability criteria (D.6) there is no restriction on the perturbations h to
the extremal, but direct analysis of (D.7) or more specifically the construction of
the projection operator P is difficult, except when the range of the tangent space
(R (∇Ci )) is simply related to the eigenspaces of the second variation of F . If
possible, one would like to avoid such computations. Here an index theory is used
as an apparatus to derive a stability criteria that does precisely that.
Before
deriving the stability result, a few remarks regarding index theory are offered. An
index theory is intimately related to catastrophe theory (Gilmore, 1981) and is
generally used to define the local character of a non-degenerate extremal. For
example, one can show that in R2 a local coordinate transformation always exists
243
(a)
(b)
(c)
Figure D.1: Local energy landscape for (a) E(0, 2) = x21 + x22 (Minimum),
(b) E(1, 1) = x21 − x22 (Saddle), and (c) E(2, 0) = −x21 − x22
(Maximum)
(a)
(b)
Low
x1
High
x2
High
x2
x2
High
(c)
Low
x1
Low
x1
Figure D.2: Contour plot of the energy landscape for (a) E(0, 2) = x21 + x22 ,
(b) E(1, 1) = x21 − x22 , and (c) E(2, 0) = −x21 − x22
which transforms the local ‘energy’ landscape of a non-degenerate extremal into
one of three unique configurations(c.f. figure D.1),
E(0, 2) = x21 + x22 ,
(D.8a)
E(1, 1) = x21 − x22 ,
(D.8b)
E(2, 0) = −x21 − x22 .
(D.8c)
These landscapes are exhaustive for non-degenerate extremals and are distinguished by the number of negative eigenvalues of their respective second variation or their index. Geometrically, E(0, 2) has no negative eigenvalue, an index
of zero and is a local minimum, E(1, 1) is saddle point with and index of one
244
(a)
EH1,1L
(b)
EH1,1L
x2
x1
Figure D.3: Projection of the energy landscape for a saddle point showing
slices of (a) constant x2 (constrained minimum) and (b) constant
x1 (constrained maximum)
and E(2, 0) is a local maximum with an index of two(c.f. figures D.1,D.2). In
either the constrained or unconstrained problem, E(0, 2) and E(2, 0) are always a
local minimum and maximum, respectively (i.e. orientation does not change the
local character). However, the saddle point E(1, 1) can be either a constrained
minimum or maximum. For example, if one holds x2 fixed the projected energy
landscape is a local minimum(stable). Alternatively, holding x1 fixed generates a
constrained maximum(unstable). These projections are shown in figure D.3. In
both such projections, one could have defined a ‘constrained’ index to describe
the local character. Here the imposed constraint fixes the ‘slice’ of the energy
landscape or the orientation from which an observer views the extremal. These
particular projections are straight lines in the two-dimensional energy landscapes
of figure D.2, but the idea of projections easily generalizes to non-linear constraints,
which are simply curves in the energy space. The geometrical interpretation of a
constrained index theory can be extended to the arbitrary Hilbert space, provided
some assumptions are made about the functional in question.
245
With regards to the constrained variational problem considered here, the number of negative eigenvalues of ∇2 F (z ∗ ) gives an ‘unconstrained’ index of the extremal of (D.2) when the null space of the second variation of F is trivial,
¡
¢
N ∇2 F (z ∗ ) = {0}.
(D.9)
Similarly, if the null space of the projected second variation is orthogonal to the
tangent space of the constraint manifold,
¡
¢
N P t ∇2 F (z ∗ )P = N (∇Ci ) , i = 1, · · · , p,
(D.10)
the number of negative eigenvalues of P t ∇2 F (z ∗ )P gives a ‘constrained’ index of
H when restricted to the constraint manifold (D.5). Although a function space is
infinite dimensional, the geometric interpretation of the constrained index from the
last paragraph still holds true. One can use the definition of the constrained index
and Lemma 1 to show that a constrained minimum necessarily has a constrained
index of zero. As before, the constrained index is not trivial to compute and to
simplify this computation, some select results from Maddocks (1985) on restricted
quadratic forms will be given.
The quadratic form associated with the linear operator L is defined as
Q(u) = (u, Lu) ,
u ∈ S ⊂ H,
(D.11)
where H is a real Hilbert space endowed with the inner product (·, ·) and S is a
dense sub-space of H. Additionally, the operator L has the following properties:
i) L is self-adjoint and Fredholm, ii) L has a finite number of negative eigenvalues
σ − with corresponding orthonormal eigenvectors ζi− , and iii) L is positive on the
orthogonal complement of span{ζi− } ⊕ N (L). The orthogonal complement of the
sub-space S is defined as
S ⊥ = { u ∈ H | (u, v) = 0 ∀ v ∈ S}.
246
(D.12)
Finally, the following decomposition of the Hilbert space H and any sub-space
thereof is needed.
Theorem 3 Let Q(u) be a quadratic form on a Hilbert space H. Then there exists
three subspaces H− , H0 , H+ such that H = H− ⊕ H0 ⊕ H+ . The three sub-spaces
are mutually orthogonal and orthogonal under the action of the operator L. Lastly,
Q(u) is negative on H− , zero on H0 and positive on H+ .
The stability criteria for the constrained variational problem in question is
now derived. Consider the quadratic form induced by the second variation of the
¡
¢
augmented Lagrangian h, ∇2 F (z ∗ )h and let the sub-space S be defined as
S = { ∇Ci (z ∗ ) , i = 1, 2, · · · , p},
(D.13)
or the tangent space to the constraint manifold. The second variation is Fredholm,
therefore the orthogonal complement to S is uniquely determined by the span of
the functions ηi defined by
S ⊥ = { ηi , i = 1, 2, · · · , p | ∇2 F (z ∗ )ηi = ∇Ci (z ∗ )}.
(D.14)
The Hilbert space is decomposed as follows
H = S ⊕ S ⊥,
(D.15)
with each sub-space decomposed as
S = S − ⊕ S 0 ⊕ S +,
¡ ¢− ¡ ¢0 ¡ ¢+
S⊥ = S⊥ ⊕ S⊥ ⊕ S⊥ .
Each subspace has its own index.
247
(D.16a)
(D.16b)
Lemma 2 The following relationship exists between the Morse indices of the
¡
¢
quadratic form h, ∇2 F (z ∗ ) h on the two complementary sub-spaces, S, S ⊥ .
£ ¤
[H] = [S] + S ⊥
(D.17)
That is, if the index on any two of the three spaces are known then the index on the
remaining space is also known. By construction, [H] is the ‘unconstrained’ index
and [S] is the ‘constrained’ index. As stated earlier, computation of the constrained
index is difficult when compared to the computation of the unconstrained index.
The goal will be to show that computing the unconstrained index and the index
on the orthogonal complement is easier than directly computing the constrained
index.
To begin, one would like to shown that the index on the orthogonal complement
can be written as
¡
´
³
¢
¡
¢−1
∇Cj (z ∗ ) . (D.18)
ηi , ∇2 F (z ∗ )ηj = (ηi , ∇Cj (z ∗ )) = ∇Ci (z ∗ ), ∇2 F (z ∗ )
This is proved by observing that any non-degenerate critical point can be embedded
in the following way: z ∗ = z ∗ (µ), which can then be applied to the first order
conditions
∗
∇H (z (µ)) =
p
X
µi ∇Ci (z ∗ (µ)).
(D.19)
i=1
Next, differentiating (D.19) with respect to µj gives
p
∂z ∗
∂z ∗ X
∇H
=
µi ∇2 Ci
+ δij ∇Ci
∂µj
∂µ
j
i=1
2
or re-arranging
Ã
2
∇ H−
p
X
!
2
µi ∇ C i
i=1
(D.20)
p
∂z ∗ X
=
δij ∇Ci .
∂µj
i=1
(D.21)
Next, using the definition of the augmented Lagrangian gives
∇2 F (z ∗ (µ))
∂z ∗
= ∇Cj (z ∗ (µ)) ,
∂µj
248
(D.22)
which allows one to explicitly construct the functions
¢−1
∂z ∗ ¡ 2
ηj (µ) =
= ∇ F (z ∗ (µ))
∇Cj (z ∗ (µ))
∂µj
(D.23)
that span the orthogonal complement, which are simply the derivatives of the
equilibria with respect to the Lagrange multipliers µj .
Finally, the previous result can be used to relate the index on the orthogonal
complement to the Hessian of the solution surface F (z(µ), µ):
Lemma 3
³
´
¡ 2
¢
∂ 2F
∗
∗
∗ −1
∗
= − (ηi , ∇Cj (z )) = ∇Ci (z ), ∇ F (z )
∇Cj (z ) .
∂µi ∂µj
(D.24)
To prove this result, differentiate the solution surface F (z(µ), µ) with respect to
µi
¶
p µ
X
∂F
∂z
∂z
= ∇H
−
δik Ck + µk ∇Ck
∂µi
∂µi k=1
∂µi
(D.25)
and then µj
!
Ã
p
X
∂ 2F
∂2z
∂z
∂z
µk ∇Ck
= ∇2 H
+ ∇H −
∂µi ∂µj
∂µi ∂µj
∂µi ∂µj
k=1
µ
¶
p
X
∂z
∂z
∂z ∂z
2
−
δik ∇Ck
+ δjk ∇Ck
+ µk ∇ Ck
.
∂µj
∂µi
∂µi ∂µj
k=1
Applying the first order conditions and simplifying yields
¶
p µ
X
∂z ∂z
∂z
∂ 2F
∂z
2
=∇ F
−
+ δjk ∇Ck
δik ∇Ck
∂µi ∂µj
∂µi ∂µj k=1
∂µj
∂µi
and finally
³
´
¡
¢−1
∂ 2F
= − ∇Ci , ∇2 F
∇Cj .
∂µi ∂µj
(D.26)
(D.27)
(D.28)
That is, the index of the orthogonal complement is simply the number of positive
eigenvalues of the Hessian of the solution surface. One can now state the stability
result:
249
Theorem 4 Given a non-degenerate extremal of F , the constrained index is the
unconstrained index minus the number of positive eigenvalues of the Hessian matrix
∂ 2 F/∂µi ∂µj .
A number of immediate results are available from Theorem 2.
Lemma 4 Suppose a non-degenerate extremal (z ∗ (µ̂), µ̂) is smoothly embedded in
a family of extremals z ∗ (µ). Then, z ∗ (µ̂) is a constrained local minimum if and
only if the number of positive principal curvatures of F (µ) equals the number of
negative eigenvalues of ∇2 F (z ∗ (µ̂)).
Lemma 5 Given p constraints and an extremal with unconstrained index l, the
extremal z ∗ is never a constrained minimum when l > p.
Finally, the class of constrained variational problem considered here is particularly suited to branch continuation. As stated earlier, computations regarding the
second variation cannot be eliminated but may be simplified. The second variation
enters Theorem 2 only through the unconstrained index, which is a much simpler
computation that the constrained index. The second requirement of Theorem 2 is
the number of positive principal curvatures of the equilibrium solution, which may
be computed solely from the first order conditions using finite differences.
D.1
Coupled non-linear springs
An example problem is used as an apparatus to illustrate the results of this subsection. Consider the static stability of the coupled non-linear spring system shown
250
Figure D.4: Schematic of the system of coupled non-linear springs
in figure D.4. The system has a configurational energy given by
U (x) =
x21
x31 x22
x23 x33
−
+
+3 − ,
3
2
4
3
(D.29)
and is subject to the following set of constraints
x1 + x2 + x3 = l1 ,
(D.30a)
x3 − x1 = l2 ,
(D.30b)
which fixes the total length l1 and the relative difference l2 between the first and last
springs. To compute the equilibria and their stability one introduces the Lagrange
multipliers µ1 , µ2 and defines the augmented Lagrangian
F (x, µ1 , µ2 ) = x21 −
x 2 x3
x31 x22
+ + 3 3 − 3 − µ1 (x1 + x2 + x3 ) − µ2 (x3 − x1 ) . (D.31)
3
2
4
3
As before, the first variation
¢
¡
∇F = 2x1 − x21 − µ1 + µ2 , x2 − µ1 , 3x3 /2 − x23 − µ1 − µ2
251
(D.32)
generates the necessary conditions for equilibrium,
x∗1 = 1 ±
p
x∗2 = µ1
1 + (µ2 − µ1 )
(D.33b)
q
(3/4)2 − (µ1 + µ2 ).
x∗3 = 3/4 ±
(D.33a)
(D.33c)
The four equilibria are parameterized by µ1 , µ2 ;
µ
p
= 1 + 1 + µ2 − µ1 ,
µ
p
∗
x 2 = 1 − 1 + µ2 − µ1 ,
µ
p
∗
x 3 = 1 + 1 + µ2 − µ1 ,
µ
p
∗
x 4 = 1 − 1 + µ2 − µ1 ,
x∗1
µ1 ,
µ1 ,
µ1 ,
µ1 ,
¶
q
2
3/4 − (3/4) − (µ1 + µ2 ) ,
¶
q
2
3/4 − (3/4) − (µ1 + µ2 ) ,
¶
q
2
3/4 + (3/4) − (µ1 + µ2 ) ,
¶
q
2
3/4 + (3/4) − (µ1 + µ2 ) .
(D.34a)
(D.34b)
(D.34c)
(D.34d)
To use Theorem 2, one needs to compute the unconstrained index for the
respective equilibria, which one can compute from the second variation


0
 2 (x1 − 1) 0



2
,
∇ F =
0
1
0




0
0 2x3 − (3/2)
(D.35)
to show that the unconstrained index of the respective equilibria are
£
¤
x∗1 = 1,
£ ∗¤
x2 = 0,
£ ∗¤
x3 = 2,
£ ∗¤
x4 = 1.
(D.36a)
(D.36b)
(D.36c)
(D.36d)
Here x∗2 is an unconstrained minimum and therefore necessarily a constrained
minimum. According to Lemma 5, the remaining equilibria have an unconstrained
252
index of l = 1, 2, and are candidates to be a constrained minima because there
are p = 2 constraints. Lastly, one computes the number of positive eigenvalues
of the Hessian matrix ∂ 2 F/∂µi ∂µj and then applies Theorem 2 to determine the
constrained index. Figure D.5 plots the solution surface for each equilibria against
the Lagrange multipliers with gray shading indicating the extremal is a constrained
minima. Projecting the solution surface onto the parameter space µ1 −µ2 generates
the stability diagram shown in figure D.6. In the stability diagram, all equilibria
with a non-trivial unconstrained index are a constrained minima in certain regions
of parameter space. Somewhat surprisingly, the equilibria with unconstrained
index of two can be a constrained minima.
253
(a)
(b)
(c)
(d)
Figure D.5: Solution surface (a) F (x∗1 ), (b) F (x∗2 ), (c) F (x∗3 ) and (d) F (x∗4 )
against Lagrange multipliers µ1 , µ2 , where gray shading indicates
a constrained minimum.
254
(a)
(b)
(c)
(d)
Figure D.6: Stability diagram for extremal (a) x∗1 , (b) x∗2 , (c) x∗3 and (d) x∗4
against Lagrange multipliers µ1 , µ2 . Gray shading indicates the
extremal is a constrained local minimum.
255
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